{"id":227058,"date":"2022-07-01T14:06:26","date_gmt":"2022-07-01T08:36:26","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=227058"},"modified":"2023-07-14T20:04:51","modified_gmt":"2023-07-14T14:34:51","slug":"quantum-physics-from-rgpv-engineering-physics-notes","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/quantum-physics-from-rgpv-engineering-physics-notes\/","title":{"rendered":"Quantum Physics &#8211; RGPV Engineering Physics Notes"},"content":{"rendered":"<\/p>\n<style type=\"text\/css\">\n<!--\n\tp {margin: 0; padding: 0;}\t.ft20{font-size:16px;font-family:Times;color:#000000;}\n\t.ft21{font-size:18px;font-family:Times;color:#000000;}\n\t.ft22{font-size:16px;font-family:Times;color:#000000;}\n\t.ft23{font-size:15px;font-family:Times;color:#000000;}\n\t.ft24{font-size:14px;font-family:Times;color:#000000;}\n\t.ft25{font-size:13px;font-family:Times;color:#000000;}\n\t.ft26{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft27{font-size:16px;line-height:24px;font-family:Times;color:#000000;}\n\t.ft28{font-size:16px;line-height:29px;font-family:Times;color:#000000;}\n\t.ft29{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft30{font-size:16px;font-family:Times;color:#000000;}\n\t.ft31{font-size:13px;font-family:Times;color:#000000;}\n\t.ft32{font-size:16px;font-family:Times;color:#000000;}\n\t.ft33{font-size:15px;font-family:Times;color:#000000;}\n\t.ft34{font-size:9px;font-family:Times;color:#000000;}\n\t.ft35{font-size:16px;line-height:27px;font-family:Times;color:#000000;}\n\t.ft36{font-size:16px;line-height:26px;font-family:Times;color:#000000;}\n\t.ft37{font-size:16px;line-height:33px;font-family:Times;color:#000000;}\n\t.ft38{font-size:16px;line-height:30px;font-family:Times;color:#000000;}\n\t.ft39{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft310{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft311{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n\t.ft312{font-size:16px;line-height:19px;font-family:Times;color:#000000;}\n\t.ft313{font-size:16px;line-height:27px;font-family:Times;color:#000000;}\n\t.ft314{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft40{font-size:16px;font-family:Times;color:#000000;}\n\t.ft41{font-size:9px;font-family:Times;color:#000000;}\n\t.ft42{font-size:15px;font-family:Times;color:#000000;}\n\t.ft43{font-size:13px;font-family:Times;color:#000000;}\n\t.ft44{font-size:12px;font-family:Times;color:#000000;}\n\t.ft45{font-size:16px;font-family:Times;color:#000000;}\n\t.ft46{font-size:-1px;font-family:Times;color:#000000;}\n\t.ft47{font-size:16px;line-height:17px;font-family:Times;color:#000000;}\n\t.ft48{font-size:16px;line-height:29px;font-family:Times;color:#000000;}\n\t.ft49{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft50{font-size:16px;font-family:Times;color:#000000;}\n\t.ft51{font-size:18px;font-family:Times;color:#000000;}\n\t.ft52{font-size:16px;font-family:Times;color:#000000;}\n\t.ft53{font-size:14px;font-family:Times;color:#000000;}\n\t.ft54{font-size:9px;font-family:Times;color:#000000;}\n\t.ft55{font-size:13px;font-family:Times;color:#000000;}\n\t.ft56{font-size:11px;font-family:Times;color:#000000;}\n\t.ft57{font-size:-1px;font-family:Times;color:#000000;}\n\t.ft58{font-size:-1px;font-family:Times;color:#ffffff;}\n\t.ft59{font-size:18px;font-family:Times;color:#000000;}\n\t.ft510{font-size:15px;font-family:Times;color:#000000;}\n\t.ft511{font-size:16px;line-height:15px;font-family:Times;color:#000000;}\n\t.ft512{font-size:16px;line-height:16px;font-family:Times;color:#000000;}\n\t.ft513{font-size:16px;line-height:27px;font-family:Times;color:#000000;}\n\t.ft514{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n\t.ft515{font-size:16px;line-height:30px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft60{font-size:16px;font-family:Times;color:#000000;}\n\t.ft61{font-size:9px;font-family:Times;color:#000000;}\n\t.ft62{font-size:13px;font-family:Times;color:#000000;}\n\t.ft63{font-size:16px;line-height:23px;font-family:Times;color:#000000;}\n\t.ft64{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft65{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n\t.ft66{font-size:16px;line-height:33px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft70{font-size:16px;font-family:Times;color:#000000;}\n\t.ft71{font-size:18px;font-family:Times;color:#000000;}\n\t.ft72{font-size:16px;font-family:Times;color:#000000;}\n\t.ft73{font-size:15px;font-family:Times;color:#000000;}\n\t.ft74{font-size:14px;font-family:Times;color:#000000;}\n\t.ft75{font-size:9px;font-family:Times;color:#000000;}\n\t.ft76{font-size:16px;line-height:30px;font-family:Times;color:#000000;}\n\t.ft77{font-size:16px;line-height:26px;font-family:Times;color:#000000;}\n\t.ft78{font-size:16px;line-height:24px;font-family:Times;color:#000000;}\n\t.ft79{font-size:16px;line-height:18px;font-family:Times;color:#000000;}\n\t.ft710{font-size:16px;line-height:21px;font-family:Times;color:#000000;}\n\t.ft711{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n\t.ft712{font-size:16px;line-height:17px;font-family:Times;color:#000000;}\n\t.ft713{font-size:16px;line-height:16px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft80{font-size:16px;font-family:Times;color:#000000;}\n\t.ft81{font-size:16px;font-family:Times;color:#000000;}\n\t.ft82{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft83{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n\t.ft84{font-size:16px;line-height:33px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft90{font-size:16px;font-family:Times;color:#000000;}\n\t.ft91{font-size:18px;font-family:Times;color:#000000;}\n\t.ft92{font-size:16px;font-family:Times;color:#000000;}\n\t.ft93{font-size:15px;font-family:Times;color:#000000;}\n\t.ft94{font-size:16px;line-height:19px;font-family:Times;color:#000000;}\n\t.ft95{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n\t.ft96{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft97{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n\t.ft98{font-size:16px;line-height:33px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft100{font-size:16px;font-family:Times;color:#000000;}\n\t.ft101{font-size:16px;font-family:Times;color:#000000;}\n\t.ft102{font-size:9px;font-family:Times;color:#000000;}\n\t.ft103{font-size:-1px;font-family:Times;color:#000000;}\n\t.ft104{font-size:-1px;font-family:Times;color:#ffffff;}\n\t.ft105{font-size:13px;font-family:Times;color:#000000;}\n\t.ft106{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n\t.ft107{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft108{font-size:16px;line-height:30px;font-family:Times;color:#000000;}\n\t.ft109{font-size:16px;line-height:23px;font-family:Times;color:#000000;}\n\t.ft1010{font-size:16px;line-height:29px;font-family:Times;color:#000000;}\n\t.ft1011{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft110{font-size:16px;font-family:Times;color:#000000;}\n\t.ft111{font-size:13px;font-family:Times;color:#000000;}\n\t.ft112{font-size:15px;font-family:Times;color:#000000;}\n\t.ft113{font-size:16px;line-height:23px;font-family:Times;color:#000000;}\n\t.ft114{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft115{font-size:16px;line-height:24px;font-family:Times;color:#000000;}\n\t.ft116{font-size:16px;line-height:30px;font-family:Times;color:#000000;}\n\t.ft117{font-size:16px;line-height:33px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft120{font-size:16px;font-family:Times;color:#000000;}\n\t.ft121{font-size:18px;font-family:Times;color:#000000;}\n\t.ft122{font-size:16px;font-family:Times;color:#000000;}\n\t.ft123{font-size:14px;font-family:Times;color:#000000;}\n\t.ft124{font-size:14px;font-family:Times;color:#000000;}\n\t.ft125{font-size:15px;font-family:Times;color:#000000;}\n\t.ft126{font-size:16px;font-family:Times;color:#000000;}\n\t.ft127{font-size:8px;font-family:Times;color:#000000;}\n\t.ft128{font-size:15px;font-family:Times;color:#000000;}\n\t.ft129{font-size:14px;line-height:29px;font-family:Times;color:#000000;}\n\t.ft1210{font-size:16px;line-height:26px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft130{font-size:16px;font-family:Times;color:#000000;}\n\t.ft131{font-size:14px;font-family:Times;color:#000000;}\n\t.ft132{font-size:16px;font-family:Times;color:#000000;}\n\t.ft133{font-size:8px;font-family:Times;color:#000000;}\n\t.ft134{font-size:15px;font-family:Times;color:#000000;}\n\t.ft135{font-size:15px;font-family:Times;color:#000000;}\n\t.ft136{font-size:17px;font-family:Times;color:#000000;}\n\t.ft137{font-size:9px;font-family:Times;color:#000000;}\n\t.ft138{font-size:14px;line-height:24px;font-family:Times;color:#000000;}\n\t.ft139{font-size:14px;line-height:23px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft140{font-size:16px;font-family:Times;color:#000000;}\n\t.ft141{font-size:14px;font-family:Times;color:#000000;}\n\t.ft142{font-size:15px;font-family:Times;color:#000000;}\n\t.ft143{font-size:9px;font-family:Times;color:#000000;}\n\t.ft144{font-size:25px;font-family:Times;color:#000000;}\n\t.ft145{font-size:16px;font-family:Times;color:#000000;}\n\t.ft146{font-size:10px;font-family:Times;color:#000000;}\n\t.ft147{font-size:8px;font-family:Times;color:#000000;}\n\t.ft148{font-size:13px;font-family:Times;color:#000000;}\n\t.ft149{font-size:16px;line-height:26px;font-family:Times;color:#000000;}\n\t.ft1410{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft150{font-size:16px;font-family:Times;color:#000000;}\n\t.ft151{font-size:18px;font-family:Times;color:#000000;}\n\t.ft152{font-size:16px;font-family:Times;color:#000000;}\n\t.ft153{font-size:14px;font-family:Times;color:#000000;}\n\t.ft154{font-size:16px;line-height:16px;font-family:Times;color:#000000;}\n\t.ft155{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft156{font-size:16px;line-height:33px;font-family:Times;color:#000000;}\n\t.ft157{font-size:16px;line-height:28px;font-family:Times;color:#000000;}\n\t.ft158{font-size:16px;line-height:25px;font-family:Times;color:#000000;}\n\t.ft159{font-size:16px;line-height:16px;font-family:Times;color:#000000;}\n\t.ft1510{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft160{font-size:16px;font-family:Times;color:#000000;}\n\t.ft161{font-size:16px;font-family:Times;color:#000000;}\n\t.ft162{font-size:13px;font-family:Times;color:#000000;}\n\t.ft163{font-size:15px;font-family:Times;color:#000000;}\n\t.ft164{font-size:9px;font-family:Times;color:#000000;}\n\t.ft165{font-size:16px;line-height:32px;font-family:Times;color:#000000;}\n\t.ft166{font-size:16px;line-height:25px;font-family:Times;color:#000000;}\n\t.ft167{font-size:16px;line-height:28px;font-family:Times;color:#000000;}\n\t.ft168{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft169{font-size:16px;line-height:18px;font-family:Times;color:#000000;}\n\t.ft1610{font-size:16px;line-height:17px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft170{font-size:16px;font-family:Times;color:#000000;}\n\t.ft171{font-size:13px;font-family:Times;color:#000000;}\n\t.ft172{font-size:16px;font-family:Times;color:#000000;}\n\t.ft173{font-size:15px;font-family:Times;color:#000000;}\n\t.ft174{font-size:10px;font-family:Times;color:#000000;}\n\t.ft175{font-size:12px;font-family:Times;color:#000000;}\n\t.ft176{font-size:16px;line-height:26px;font-family:Times;color:#000000;}\n\t.ft177{font-size:16px;line-height:17px;font-family:Times;color:#000000;}\n\t.ft178{font-size:16px;line-height:23px;font-family:Times;color:#000000;}\n\t.ft179{font-size:16px;line-height:22px;font-family:Times;color:#000000;}\n\t.ft1710{font-size:16px;line-height:29px;font-family:Times;color:#000000;}\n\t.ft1711{font-size:16px;line-height:30px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft180{font-size:16px;font-family:Times;color:#000000;}\n\t.ft181{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft190{font-size:16px;font-family:Times;color:#000000;}\n\t.ft191{font-size:18px;font-family:Times;color:#000000;}\n\t.ft192{font-size:16px;font-family:Times;color:#000000;}\n\t.ft193{font-size:15px;font-family:Times;color:#000000;}\n\t.ft194{font-size:13px;font-family:Times;color:#000000;}\n\t.ft195{font-size:10px;font-family:Times;color:#000000;}\n\t.ft196{font-size:12px;font-family:Times;color:#000000;}\n\t.ft197{font-size:14px;font-family:Times;color:#000000;}\n\t.ft198{font-size:16px;line-height:30px;font-family:Times;color:#000000;}\n\t.ft199{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft200{font-size:16px;font-family:Times;color:#000000;}\n\t.ft201{font-size:14px;font-family:Times;color:#000000;}\n\t.ft202{font-size:15px;font-family:Times;color:#000000;}\n\t.ft203{font-size:13px;font-family:Times;color:#000000;}\n\t.ft204{font-size:9px;font-family:Times;color:#000000;}\n\t.ft205{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n\t.ft206{font-size:16px;line-height:18px;font-family:Times;color:#000000;}\n\t.ft207{font-size:16px;line-height:25px;font-family:Times;color:#000000;}\n\t.ft208{font-size:16px;line-height:24px;font-family:Times;color:#000000;}\n\t.ft209{font-size:16px;line-height:19px;font-family:Times;color:#000000;}\n\t.ft2010{font-size:16px;line-height:27px;font-family:Times;color:#000000;}\n\t.ft2011{font-size:16px;line-height:17px;font-family:Times;color:#000000;}\n\t.ft2012{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft210{font-size:16px;font-family:Times;color:#000000;}\n\t.ft211{font-size:15px;font-family:Times;color:#000000;}\n\t.ft212{font-size:9px;font-family:Times;color:#000000;}\n\t.ft213{font-size:12px;font-family:Times;color:#000000;}\n\t.ft214{font-size:13px;font-family:Times;color:#000000;}\n\t.ft215{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n\t.ft216{font-size:16px;line-height:31px;font-family:Times;color:#000000;}\n\t.ft217{font-size:16px;line-height:28px;font-family:Times;color:#000000;}\n\t.ft218{font-size:16px;line-height:16px;font-family:Times;color:#000000;}\n\t.ft219{font-size:16px;line-height:30px;font-family:Times;color:#000000;}\n\t.ft2110{font-size:16px;line-height:23px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft220{font-size:16px;font-family:Times;color:#000000;}\n\t.ft221{font-size:15px;font-family:Times;color:#000000;}\n\t.ft222{font-size:14px;font-family:Times;color:#000000;}\n\t.ft223{font-size:18px;font-family:Times;color:#000000;}\n\t.ft224{font-size:16px;font-family:Times;color:#000000;}\n\t.ft225{font-size:9px;font-family:Times;color:#000000;}\n\t.ft226{font-size:13px;font-family:Times;color:#000000;}\n\t.ft227{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n\t.ft228{font-size:16px;line-height:29px;font-family:Times;color:#000000;}\n\t.ft229{font-size:16px;line-height:19px;font-family:Times;color:#000000;}\n\t.ft2210{font-size:16px;line-height:15px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft230{font-size:16px;font-family:Times;color:#000000;}\n\t.ft231{font-size:13px;font-family:Times;color:#000000;}\n\t.ft232{font-size:9px;font-family:Times;color:#000000;}\n\t.ft233{font-size:16px;font-family:Times;color:#000000;}\n\t.ft234{font-size:15px;font-family:Times;color:#000000;}\n\t.ft235{font-size:16px;line-height:15px;font-family:Times;color:#000000;}\n\t.ft236{font-size:16px;line-height:19px;font-family:Times;color:#000000;}\n\t.ft237{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n--><\/p>\n<p><!--\n\tp {margin: 0; padding: 0;}\t.ft240{font-size:16px;font-family:Times;color:#000000;}\n\t.ft241{font-size:18px;font-family:Times;color:#000000;}\n\t.ft242{font-size:16px;font-family:Times;color:#000000;}\n\t.ft243{font-size:14px;font-family:Times;color:#000000;}\n\t.ft244{font-size:16px;line-height:19px;font-family:Times;color:#000000;}\n\t.ft245{font-size:16px;line-height:20px;font-family:Times;color:#000000;}\n\t.ft246{font-size:16px;line-height:27px;font-family:Times;color:#000000;}\n-->\n<\/style>\n<div id=\"page2-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224002.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft20\">1&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:443px;white-space:nowrap\" class=\"ft21\"><b>UNIT&nbsp;&ndash;&nbsp;1&nbsp;&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:85px;left:150px;white-space:nowrap\" class=\"ft21\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:110px;left:393px;white-space:nowrap\" class=\"ft21\"><b>QUANTUM&nbsp;PHYSICS&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:150px;left:397px;white-space:nowrap\" class=\"ft21\"><b>Unit-01\/Lecture-01&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:189px;left:150px;white-space:nowrap\" class=\"ft22\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:222px;left:150px;white-space:nowrap\" class=\"ft21\"><b>Concept&nbsp;of&nbsp;matter&nbsp;waves&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:259px;left:150px;white-space:nowrap\" class=\"ft26\">Louis&nbsp;de&nbsp;Broglie&nbsp;made&nbsp;the&nbsp;suggestion that particles&nbsp;of matter,&nbsp;like&nbsp;electrons, might possess&nbsp;<br \/>wave&nbsp;properties&nbsp;and hence&nbsp;exhibit dual&nbsp;nature.&nbsp;His&nbsp;hypothesis&nbsp;was&nbsp;based on the&nbsp;following&nbsp;<br \/>arguments:&nbsp;<\/p>\n<p style=\"position:absolute;top:340px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:358px;left:150px;white-space:nowrap\" class=\"ft20\">The&nbsp;Planck&rsquo;s&nbsp; theory&nbsp; of &nbsp;radiation &nbsp;suggests&nbsp; that &nbsp;energy&nbsp;<\/p>\n<p style=\"position:absolute;top:371px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:390px;left:150px;white-space:nowrap\" class=\"ft20\">is&nbsp;quantized and is&nbsp;given&nbsp;by&nbsp;<\/p>\n<p style=\"position:absolute;top:410px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:431px;left:214px;white-space:nowrap\" class=\"ft20\">E = h<\/p>\n<p style=\"position:absolute;top:425px;left:268px;white-space:nowrap\" class=\"ft20\">&nu;&nbsp;<\/p>\n<p style=\"position:absolute;top:432px;left:676px;white-space:nowrap\" class=\"ft23\">(1)<\/p>\n<p style=\"position:absolute;top:430px;left:707px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:444px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:458px;left:150px;white-space:nowrap\" class=\"ft20\">where&nbsp;&nu;&nbsp;is&nbsp;the&nbsp;frequency&nbsp;associated&nbsp;with the&nbsp;radiation.&nbsp;<\/p>\n<p style=\"position:absolute;top:480px;left:150px;white-space:nowrap\" class=\"ft27\">Einstein&rsquo;s&nbsp;mass-energy&nbsp;relation states&nbsp;that&nbsp;<br \/>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:513px;left:239px;white-space:nowrap\" class=\"ft24\">E = mc<\/p>\n<p style=\"position:absolute;top:503px;left:299px;white-space:nowrap\" class=\"ft25\">2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2)<\/p>\n<p style=\"position:absolute;top:505px;left:705px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:529px;left:150px;white-space:nowrap\" class=\"ft24\">Combining&nbsp;the two&nbsp;equations,&nbsp;it&nbsp;can&nbsp;be&nbsp;written&nbsp;as<\/p>\n<p style=\"position:absolute;top:528px;left:483px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:561px;left:214px;white-space:nowrap\" class=\"ft24\">E = h<\/p>\n<p style=\"position:absolute;top:554px;left:264px;white-space:nowrap\" class=\"ft24\">&nu;&nbsp;=&nbsp;mc<\/p>\n<p style=\"position:absolute;top:550px;left:322px;white-space:nowrap\" class=\"ft25\">2&nbsp;<\/p>\n<p style=\"position:absolute;top:577px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:599px;left:150px;white-space:nowrap\" class=\"ft20\">Hence, the&nbsp;momentum&nbsp;associated with the&nbsp;photon is&nbsp;given&nbsp;by&nbsp;<\/p>\n<p style=\"position:absolute;top:615px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:636px;left:214px;white-space:nowrap\" class=\"ft20\">P = mc = h<\/p>\n<p style=\"position:absolute;top:629px;left:322px;white-space:nowrap\" class=\"ft20\">&nu;\/c = h\/&lambda;&nbsp;<\/p>\n<p style=\"position:absolute;top:653px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:667px;left:150px;white-space:nowrap\" class=\"ft28\">Extending&nbsp;this&nbsp;to&nbsp;particles, he&nbsp;suggested that any&nbsp;particle&nbsp;having&nbsp;a&nbsp;&nbsp;momentum p&nbsp;is&nbsp;<br \/>associated with a&nbsp;wave&nbsp;of wavelength given by&nbsp;<\/p>\n<p style=\"position:absolute;top:719px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:732px;left:214px;white-space:nowrap\" class=\"ft20\">&lambda;&nbsp;= h\/p&nbsp;<\/p>\n<p style=\"position:absolute;top:739px;left:711px;white-space:nowrap\" class=\"ft20\">(3)<\/p>\n<p style=\"position:absolute;top:738px;left:744px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:755px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:769px;left:150px;white-space:nowrap\" class=\"ft20\">This&nbsp;is&nbsp;called&nbsp;<\/p>\n<p style=\"position:absolute;top:769px;left:257px;white-space:nowrap\" class=\"ft22\"><b>de<\/b>&nbsp;&nbsp;<b>Broglie&rsquo;s hypothesis<\/b>&nbsp;&nbsp;of&nbsp;matter&nbsp;waves&nbsp;and&nbsp;<\/p>\n<p style=\"position:absolute;top:771px;left:613px;white-space:nowrap\" class=\"ft24\">&lambda;<\/p>\n<p style=\"position:absolute;top:776px;left:622px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:769px;left:630px;white-space:nowrap\" class=\"ft20\">is&nbsp;called&nbsp;the&nbsp;de&nbsp;Broglie&nbsp;<\/p>\n<p style=\"position:absolute;top:797px;left:150px;white-space:nowrap\" class=\"ft29\">wavelength.&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:849px;left:193px;white-space:nowrap\" class=\"ft20\">In&nbsp;case&nbsp;of&nbsp;charged&nbsp;particles&nbsp;like&nbsp;electrons,&nbsp;a beam of&nbsp;high&nbsp;energy&nbsp;particles&nbsp;can&nbsp;be&nbsp;<\/p>\n<p style=\"position:absolute;top:881px;left:150px;white-space:nowrap\" class=\"ft26\">obtained&nbsp;by&nbsp;accelerating&nbsp;them in&nbsp;an&nbsp;electric&nbsp;field.&nbsp;For example,&nbsp;an&nbsp;electron&nbsp;starting&nbsp;from&nbsp;<br \/>rest when accelerated&nbsp;with a&nbsp;potential&nbsp;difference&nbsp;V, the&nbsp;kinetic&nbsp;energy&nbsp;acquired&nbsp;by&nbsp;the&nbsp;<br \/>electron is&nbsp;given by&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:999px;left:225px;white-space:nowrap\" class=\"ft20\">(1\/2)mv<\/p>\n<p style=\"position:absolute;top:990px;left:300px;white-space:nowrap\" class=\"ft25\">2<\/p>\n<p style=\"position:absolute;top:999px;left:310px;white-space:nowrap\" class=\"ft20\">&nbsp;&nbsp;= eV<\/p>\n<p style=\"position:absolute;top:998px;left:375px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1007px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1021px;left:150px;white-space:nowrap\" class=\"ft25\">where&nbsp;&nbsp;v &nbsp;is&nbsp;&nbsp;the&nbsp;&nbsp;velocity&nbsp; of&nbsp;&nbsp;the&nbsp;&nbsp;electron.&nbsp; The&nbsp;&nbsp;momentum<\/p>\n<p style=\"position:absolute;top:1019px;left:535px;white-space:nowrap\" class=\"ft20\">&nbsp;may&nbsp;be&nbsp;calculated&nbsp;as&nbsp;<\/p>\n<p style=\"position:absolute;top:1034px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1064px;left:225px;white-space:nowrap\" class=\"ft20\">p = mv = (2meV)<\/p>\n<p style=\"position:absolute;top:1055px;left:387px;white-space:nowrap\" class=\"ft25\">1\/2<\/p>\n<p style=\"position:absolute;top:1063px;left:415px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1071px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1084px;left:150px;white-space:nowrap\" class=\"ft23\">Using<\/p>\n<p style=\"position:absolute;top:1083px;left:189px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1084px;left:194px;white-space:nowrap\" class=\"ft23\">the&nbsp;de&nbsp;Broglie&nbsp;equation,&nbsp;the&nbsp;wavelength<\/p>\n<p style=\"position:absolute;top:1083px;left:477px;white-space:nowrap\" class=\"ft20\">&nbsp;associated with the&nbsp;accelerated electron can&nbsp;<\/p>\n<p style=\"position:absolute;top:1105px;left:150px;white-space:nowrap\" class=\"ft20\">be&nbsp;calculated&nbsp;as&nbsp;<\/p>\n<p style=\"position:absolute;top:1122px;left:150px;white-space:nowrap\" class=\"ft20\">&nbsp;<\/p>\n<\/div>\n<div id=\"page3-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224003.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft30\">2&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:53px;left:225px;white-space:nowrap\" class=\"ft30\">&lambda;&nbsp;= h\/p = h\/(2meV)<\/p>\n<p style=\"position:absolute;top:51px;left:418px;white-space:nowrap\" class=\"ft31\">1\/2<\/p>\n<p style=\"position:absolute;top:59px;left:447px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:62px;left:711px;white-space:nowrap\" class=\"ft30\">(4)<\/p>\n<p style=\"position:absolute;top:61px;left:744px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:72px;left:150px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:84px;left:150px;white-space:nowrap\" class=\"ft37\">This&nbsp;equation suggests&nbsp;that, at a&nbsp;given speed, the&nbsp;de&nbsp;Broglie&nbsp;wavelength&nbsp;associated&nbsp;with&nbsp;<br \/>the&nbsp;particle&nbsp;varies&nbsp;inversely&nbsp;as&nbsp;the&nbsp;mass&nbsp;of&nbsp;the&nbsp;particle.&nbsp;<br \/><b>&nbsp;<br \/><\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:171px;left:154px;white-space:nowrap\" class=\"ft32\"><b>Definitions&nbsp;[Rgpv&nbsp;June&nbsp;2011,&nbsp;Dec&nbsp;2011&nbsp;(7)]&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:198px;left:150px;white-space:nowrap\" class=\"ft35\"><b>Wave&nbsp;Packet<\/b>&nbsp;<br \/>A&nbsp;<\/p>\n<p style=\"position:absolute;top:226px;left:167px;white-space:nowrap\" class=\"ft32\"><b>wave packet<\/b>&nbsp;&nbsp;consisting&nbsp;of&nbsp;waves&nbsp;of&nbsp;slightly&nbsp;differing&nbsp;wavelengths&nbsp;may&nbsp;represent the&nbsp;<\/p>\n<p style=\"position:absolute;top:257px;left:150px;white-space:nowrap\" class=\"ft30\">moving&nbsp;particle.&nbsp;Superposition of these&nbsp;waves&nbsp;constituting&nbsp;the&nbsp;wave&nbsp;packet results&nbsp;in&nbsp;the&nbsp;<\/p>\n<p style=\"position:absolute;top:289px;left:150px;white-space:nowrap\" class=\"ft33\">net&nbsp;amplitude&nbsp; being&nbsp; modified,&nbsp; thereby&nbsp; defining&nbsp; the<\/p>\n<p style=\"position:absolute;top:288px;left:528px;white-space:nowrap\" class=\"ft30\">&nbsp;shape&nbsp;of the&nbsp;wave&nbsp;group.&nbsp;<\/p>\n<p style=\"position:absolute;top:319px;left:150px;white-space:nowrap\" class=\"ft313\">&nbsp;<br \/><b>Phase velocity&nbsp;<br \/><\/b>The&nbsp;velocity&nbsp;of&nbsp;a individual wave&nbsp;of&nbsp;a wave&nbsp;packet&nbsp;is&nbsp;known&nbsp;as&nbsp;Phase&nbsp;velocity.&nbsp;<br \/>&nbsp;<br \/><b>Group velocity&nbsp;<br \/><\/b>Group velocity&nbsp;is&nbsp;the&nbsp;velocity&nbsp;with which the&nbsp;wave&nbsp;packet travels.&nbsp;<br \/>&nbsp;<br \/><b>Q.&nbsp;Derive&nbsp;the formula&nbsp;of&nbsp;Phase&nbsp;velocity&nbsp;and&nbsp;Group&nbsp;velocity&nbsp;and&nbsp;also find&nbsp;relation&nbsp;<br \/>between&nbsp;them?&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:590px;left:183px;white-space:nowrap\" class=\"ft30\">A&nbsp;wave is&nbsp;represented&nbsp;by the formula&nbsp;<\/p>\n<p style=\"position:absolute;top:590px;left:625px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:631px;left:204px;white-space:nowrap\" class=\"ft30\">y = A cos (<\/p>\n<p style=\"position:absolute;top:625px;left:323px;white-space:nowrap\" class=\"ft30\">&omega;t&nbsp;&ndash;&nbsp;kx)&nbsp;<\/p>\n<p style=\"position:absolute;top:631px;left:713px;white-space:nowrap\" class=\"ft30\">(1)<\/p>\n<p style=\"position:absolute;top:630px;left:745px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:644px;left:150px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:657px;left:150px;white-space:nowrap\" class=\"ft30\">where&nbsp;y&nbsp;is&nbsp;the&nbsp;displacement at any&nbsp;instant t, A&nbsp;is&nbsp;the&nbsp;<\/p>\n<p style=\"position:absolute;top:658px;left:534px;white-space:nowrap\" class=\"ft33\">amplitude&nbsp; of&nbsp; vibration,&nbsp;&nbsp;&omega;&nbsp;&nbsp;is&nbsp; the&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:681px;left:150px;white-space:nowrap\" class=\"ft33\">angular &nbsp;frequency<\/p>\n<p style=\"position:absolute;top:680px;left:281px;white-space:nowrap\" class=\"ft30\">&nbsp;equal&nbsp;to&nbsp;2&pi;&nu;&nbsp;and k is&nbsp;the&nbsp;wave&nbsp;vector, equal&nbsp;to&nbsp;(2&pi;\/&lambda;).&nbsp;The&nbsp;phase&nbsp;<\/p>\n<p style=\"position:absolute;top:703px;left:150px;white-space:nowrap\" class=\"ft30\">velocity&nbsp;of such a&nbsp;wave&nbsp;is&nbsp;the&nbsp;velocity&nbsp;with which a&nbsp;particular&nbsp;phase&nbsp;point of the&nbsp;wave&nbsp;<\/p>\n<p style=\"position:absolute;top:725px;left:150px;white-space:nowrap\" class=\"ft38\">travels.&nbsp;<br \/>This&nbsp;corresponds&nbsp;to&nbsp;the&nbsp;phase&nbsp;being&nbsp;constant.&nbsp;<\/p>\n<p style=\"position:absolute;top:762px;left:655px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:789px;left:150px;white-space:nowrap\" class=\"ft30\">i.e.,&nbsp;(&omega;t&nbsp;&ndash;&nbsp;kx) =&nbsp;constant&nbsp;<\/p>\n<p style=\"position:absolute;top:796px;left:655px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:831px;left:150px;white-space:nowrap\" class=\"ft30\">or<\/p>\n<p style=\"position:absolute;top:830px;left:171px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:830px;left:237px;white-space:nowrap\" class=\"ft30\">x =&nbsp;<\/p>\n<p style=\"position:absolute;top:824px;left:265px;white-space:nowrap\" class=\"ft30\">constant&nbsp;+&nbsp;&omega;t\/k&nbsp;<\/p>\n<p style=\"position:absolute;top:831px;left:655px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:858px;left:150px;white-space:nowrap\" class=\"ft30\">Phase&nbsp;velocity&nbsp;v<\/p>\n<p style=\"position:absolute;top:869px;left:267px;white-space:nowrap\" class=\"ft34\">p<\/p>\n<p style=\"position:absolute;top:865px;left:273px;white-space:nowrap\" class=\"ft30\">&nbsp;&nbsp;= dx\/dt = &nbsp;<\/p>\n<p style=\"position:absolute;top:859px;left:414px;white-space:nowrap\" class=\"ft30\">&omega;\/k&nbsp;<\/p>\n<p style=\"position:absolute;top:865px;left:655px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:900px;left:150px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:900px;left:345px;white-space:nowrap\" class=\"ft30\">= 2<\/p>\n<p style=\"position:absolute;top:893px;left:377px;white-space:nowrap\" class=\"ft30\">&pi;&nu;\/(2&pi;\/&lambda;) =&nbsp;&lambda;&nu;&nbsp;<\/p>\n<p style=\"position:absolute;top:900px;left:715px;white-space:nowrap\" class=\"ft30\">(2)<\/p>\n<p style=\"position:absolute;top:899px;left:747px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:916px;left:150px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:930px;left:150px;white-space:nowrap\" class=\"ft30\">v<\/p>\n<p style=\"position:absolute;top:937px;left:158px;white-space:nowrap\" class=\"ft34\">p<\/p>\n<p style=\"position:absolute;top:930px;left:164px;white-space:nowrap\" class=\"ft30\">&nbsp;is&nbsp;called&nbsp;the&nbsp;&lsquo;wave&nbsp;velocity&rsquo;&nbsp;or&nbsp;<\/p>\n<p style=\"position:absolute;top:930px;left:391px;white-space:nowrap\" class=\"ft32\"><b>&lsquo;phase velocity&rsquo;.&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:958px;left:150px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:980px;left:150px;white-space:nowrap\" class=\"ft30\">For&nbsp;group velocity, consider&nbsp;the&nbsp;combination of two&nbsp;waves&nbsp;represented by&nbsp;the&nbsp;<\/p>\n<p style=\"position:absolute;top:1002px;left:150px;white-space:nowrap\" class=\"ft30\">formula&nbsp;<\/p>\n<p style=\"position:absolute;top:988px;left:811px;white-space:nowrap\" class=\"ft314\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:1032px;left:150px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1039px;left:769px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1052px;left:204px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1073px;left:237px;white-space:nowrap\" class=\"ft30\">y<\/p>\n<p style=\"position:absolute;top:1077px;left:248px;white-space:nowrap\" class=\"ft34\">1<\/p>\n<p style=\"position:absolute;top:1073px;left:255px;white-space:nowrap\" class=\"ft30\">&nbsp;&nbsp;= A cos (<\/p>\n<p style=\"position:absolute;top:1067px;left:374px;white-space:nowrap\" class=\"ft30\">&omega;t-kx)&nbsp;<\/p>\n<p style=\"position:absolute;top:1092px;left:204px;white-space:nowrap\" class=\"ft30\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1115px;left:236px;white-space:nowrap\" class=\"ft30\">y<\/p>\n<p style=\"position:absolute;top:1119px;left:247px;white-space:nowrap\" class=\"ft34\">2<\/p>\n<p style=\"position:absolute;top:1115px;left:254px;white-space:nowrap\" class=\"ft30\">&nbsp;= A cos {(<\/p>\n<p style=\"position:absolute;top:1108px;left:373px;white-space:nowrap\" class=\"ft30\">&omega;+&#8710;&omega;)t&nbsp;&ndash;&nbsp;(k+&#8710;k)x}&nbsp;<\/p>\n<\/div>\n<div id=\"page4-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224004.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft40\">3&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:236px;white-space:nowrap\" class=\"ft40\">The&nbsp;resultant displacement is&nbsp;given by&nbsp;<\/p>\n<p style=\"position:absolute;top:66px;left:204px;white-space:nowrap\" class=\"ft47\">&nbsp;<br \/>y<\/p>\n<p style=\"position:absolute;top:82px;left:215px;white-space:nowrap\" class=\"ft40\">&nbsp;&nbsp;= y<\/p>\n<p style=\"position:absolute;top:88px;left:257px;white-space:nowrap\" class=\"ft41\">1<\/p>\n<p style=\"position:absolute;top:84px;left:264px;white-space:nowrap\" class=\"ft40\">&nbsp;&nbsp;+ y<\/p>\n<p style=\"position:absolute;top:88px;left:318px;white-space:nowrap\" class=\"ft41\">2<\/p>\n<p style=\"position:absolute;top:84px;left:326px;white-space:nowrap\" class=\"ft40\">&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:108px;left:161px;white-space:nowrap\" class=\"ft40\">=<\/p>\n<p style=\"position:absolute;top:106px;left:171px;white-space:nowrap\" class=\"ft40\">&nbsp;&nbsp;2A cos {(<\/p>\n<p style=\"position:absolute;top:101px;left:280px;white-space:nowrap\" class=\"ft40\">&omega;+&omega;+&#8710;&omega;)t&ndash;(k+k+&#8710;k)x} cos&nbsp;(&#8710;&omega;t-&#8710;kx)&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:128px;left:375px;white-space:nowrap\" class=\"ft40\">2<\/p>\n<p style=\"position:absolute;top:127px;left:385px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:128px;left:591px;white-space:nowrap\" class=\"ft40\">2<\/p>\n<p style=\"position:absolute;top:127px;left:601px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:144px;left:173px;white-space:nowrap\" class=\"ft40\">&asymp;&nbsp;2A&nbsp;cos(&omega;t&ndash;kx).cos(&#8710;&omega;t\/2-&#8710;kx\/2)&nbsp;<\/p>\n<p style=\"position:absolute;top:172px;left:517px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:172px;left:704px;white-space:nowrap\" class=\"ft40\">(3)<\/p>\n<p style=\"position:absolute;top:171px;left:736px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:187px;left:150px;white-space:nowrap\" class=\"ft40\">The&nbsp;&nbsp;velocity&nbsp; of &nbsp;the&nbsp; resultant &nbsp;wave&nbsp; is&nbsp; given &nbsp;by&nbsp;the&nbsp;speed with which a&nbsp;reference&nbsp;<\/p>\n<p style=\"position:absolute;top:209px;left:150px;white-space:nowrap\" class=\"ft40\">point, say&nbsp;the&nbsp;maximum&nbsp;amplitude&nbsp;point, moves.&nbsp;Taking&nbsp;the&nbsp;amplitude&nbsp;of the&nbsp;resultant&nbsp;<\/p>\n<p style=\"position:absolute;top:231px;left:150px;white-space:nowrap\" class=\"ft40\">wave&nbsp;as&nbsp;constant,&nbsp;<\/p>\n<p style=\"position:absolute;top:261px;left:193px;white-space:nowrap\" class=\"ft40\">2A cos(<\/p>\n<p style=\"position:absolute;top:254px;left:269px;white-space:nowrap\" class=\"ft40\">&#8710;&omega;t\/2-&#8710;kx\/2) =&nbsp;constant&nbsp;<\/p>\n<p style=\"position:absolute;top:282px;left:150px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:306px;left:207px;white-space:nowrap\" class=\"ft40\">or<\/p>\n<p style=\"position:absolute;top:305px;left:228px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:306px;left:264px;white-space:nowrap\" class=\"ft40\">(<\/p>\n<p style=\"position:absolute;top:299px;left:275px;white-space:nowrap\" class=\"ft40\">&#8710;&omega;t\/2-&#8710;kx\/2) =&nbsp;constant&nbsp;<\/p>\n<p style=\"position:absolute;top:327px;left:150px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:350px;left:204px;white-space:nowrap\" class=\"ft40\">or x =&nbsp;<\/p>\n<p style=\"position:absolute;top:349px;left:279px;white-space:nowrap\" class=\"ft40\">constant&nbsp;+ (<\/p>\n<p style=\"position:absolute;top:343px;left:383px;white-space:nowrap\" class=\"ft40\">&#8710;&omega;t\/&#8710;k)&nbsp;<\/p>\n<p style=\"position:absolute;top:363px;left:150px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:378px;left:160px;white-space:nowrap\" class=\"ft40\">Group velocity&nbsp;v<\/p>\n<p style=\"position:absolute;top:384px;left:279px;white-space:nowrap\" class=\"ft41\">g<\/p>\n<p style=\"position:absolute;top:384px;left:284px;white-space:nowrap\" class=\"ft40\">&nbsp;&nbsp;= dx\/dt = (<\/p>\n<p style=\"position:absolute;top:378px;left:419px;white-space:nowrap\" class=\"ft40\">&#8710;&omega;\/&#8710;k)&nbsp;<\/p>\n<p style=\"position:absolute;top:385px;left:676px;white-space:nowrap\" class=\"ft42\">(4)<\/p>\n<p style=\"position:absolute;top:384px;left:707px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:404px;left:150px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:419px;left:258px;white-space:nowrap\" class=\"ft40\">Instead &nbsp;of &nbsp;two&nbsp; discrete&nbsp; values&nbsp; for<\/p>\n<p style=\"position:absolute;top:428px;left:520px;white-space:nowrap\" class=\"ft43\">&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:422px;left:539px;white-space:nowrap\" class=\"ft43\">&omega;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:421px;left:569px;white-space:nowrap\" class=\"ft43\">and&nbsp;&nbsp;k,<\/p>\n<p style=\"position:absolute;top:425px;left:625px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:444px;left:150px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:460px;left:150px;white-space:nowrap\" class=\"ft40\">if the&nbsp;group of waves&nbsp;has&nbsp;a&nbsp;continuous&nbsp;spread from&nbsp;&nbsp;&omega;&nbsp;&nbsp;to&nbsp;&nbsp;(&omega;+&#8710;&omega;)&nbsp;&nbsp;and&nbsp;&nbsp;k&nbsp;&nbsp;to&nbsp;<\/p>\n<p style=\"position:absolute;top:498px;left:150px;white-space:nowrap\" class=\"ft40\">(k+<\/p>\n<p style=\"position:absolute;top:491px;left:182px;white-space:nowrap\" class=\"ft40\">&#8710;k),&nbsp;<\/p>\n<p style=\"position:absolute;top:491px;left:236px;white-space:nowrap\" class=\"ft40\">then, the&nbsp;group velocity&nbsp;is&nbsp;given by&nbsp;<\/p>\n<p style=\"position:absolute;top:506px;left:150px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:532px;left:204px;white-space:nowrap\" class=\"ft40\">v<\/p>\n<p style=\"position:absolute;top:536px;left:215px;white-space:nowrap\" class=\"ft41\">g<\/p>\n<p style=\"position:absolute;top:532px;left:222px;white-space:nowrap\" class=\"ft40\">&nbsp;= d<\/p>\n<p style=\"position:absolute;top:525px;left:265px;white-space:nowrap\" class=\"ft40\">&omega;&nbsp;<\/p>\n<p style=\"position:absolute;top:532px;left:717px;white-space:nowrap\" class=\"ft40\">(5)<\/p>\n<p style=\"position:absolute;top:531px;left:750px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:553px;left:258px;white-space:nowrap\" class=\"ft40\">d<\/p>\n<p style=\"position:absolute;top:555px;left:269px;white-space:nowrap\" class=\"ft44\">k<\/p>\n<p style=\"position:absolute;top:552px;left:276px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:553px;left:480px;white-space:nowrap\" class=\"ft42\">&nbsp;<\/p>\n<p style=\"position:absolute;top:561px;left:150px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:573px;left:150px;white-space:nowrap\" class=\"ft49\">It can be&nbsp;shown that the&nbsp;group velocity&nbsp;of the&nbsp;wave&nbsp;packet&nbsp;is&nbsp;equal&nbsp;to&nbsp;the&nbsp;velocity&nbsp;of the&nbsp;<br \/>particle&nbsp;with which the&nbsp;wave&nbsp;packet&nbsp;is&nbsp;associated.&nbsp;<br \/>&nbsp;<br \/><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:688px;left:158px;white-space:nowrap\" class=\"ft40\">S.NO&nbsp;<\/p>\n<p style=\"position:absolute;top:688px;left:342px;white-space:nowrap\" class=\"ft40\">RGPV&nbsp;QUESTIONS&nbsp;<\/p>\n<p style=\"position:absolute;top:688px;left:639px;white-space:nowrap\" class=\"ft40\">Year&nbsp;<\/p>\n<p style=\"position:absolute;top:688px;left:738px;white-space:nowrap\" class=\"ft40\">Marks&nbsp;<\/p>\n<p style=\"position:absolute;top:711px;left:158px;white-space:nowrap\" class=\"ft40\">Q.1&nbsp;<\/p>\n<p style=\"position:absolute;top:711px;left:225px;white-space:nowrap\" class=\"ft40\">What is&nbsp;eave&nbsp;packet?&nbsp;Define&nbsp;group velocity&nbsp;and&nbsp;<\/p>\n<p style=\"position:absolute;top:733px;left:225px;white-space:nowrap\" class=\"ft40\">phase&nbsp;velocity.&nbsp;Derive&nbsp;an&nbsp;expression for&nbsp;the&nbsp;de&nbsp;<\/p>\n<p style=\"position:absolute;top:755px;left:225px;white-space:nowrap\" class=\"ft40\">Broglie&nbsp;wavelength associated with&nbsp;an electron&nbsp;<\/p>\n<p style=\"position:absolute;top:777px;left:225px;white-space:nowrap\" class=\"ft40\">accelerated&nbsp;by the electric potential&nbsp;V.&nbsp;<\/p>\n<p style=\"position:absolute;top:711px;left:621px;white-space:nowrap\" class=\"ft40\">Dec&nbsp;2011&nbsp;<\/p>\n<p style=\"position:absolute;top:711px;left:752px;white-space:nowrap\" class=\"ft40\">14&nbsp;<\/p>\n<p style=\"position:absolute;top:799px;left:158px;white-space:nowrap\" class=\"ft40\">Q.2&nbsp;<\/p>\n<p style=\"position:absolute;top:799px;left:225px;white-space:nowrap\" class=\"ft40\">Derive&nbsp;an expression for&nbsp;the&nbsp;group velocity&nbsp;and&nbsp;<\/p>\n<p style=\"position:absolute;top:821px;left:225px;white-space:nowrap\" class=\"ft40\">phase&nbsp;velocity.&nbsp;Also&nbsp;find&nbsp;relation between them.&nbsp;<\/p>\n<p style=\"position:absolute;top:799px;left:618px;white-space:nowrap\" class=\"ft40\">June&nbsp;2011&nbsp;<\/p>\n<p style=\"position:absolute;top:799px;left:752px;white-space:nowrap\" class=\"ft40\">14&nbsp;<\/p>\n<p style=\"position:absolute;top:845px;left:150px;white-space:nowrap\" class=\"ft46\">&nbsp;<\/p>\n<p style=\"position:absolute;top:845px;left:459px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:867px;left:108px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<p style=\"position:absolute;top:867px;left:324px;white-space:nowrap\" class=\"ft40\">&nbsp;<\/p>\n<\/div>\n<div id=\"page5-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224005.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft50\">4&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:400px;white-space:nowrap\" class=\"ft51\"><b>Unit-01\/Lecture-02&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:84px;left:483px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:107px;left:157px;white-space:nowrap\" class=\"ft52\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:142px;left:157px;white-space:nowrap\" class=\"ft53\"><b>Relation&nbsp; &nbsp;between&nbsp; &nbsp;phase&nbsp; &nbsp;velocity&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;group<\/b><\/p>\n<p style=\"position:absolute;top:140px;left:486px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:140px;left:494px;white-space:nowrap\" class=\"ft52\"><b>velocity:&nbsp;[Rgpv&nbsp;June&nbsp;2013(7)]<\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:170px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;The&nbsp;mathematical relation&nbsp;for phase&nbsp;velocity&nbsp;given&nbsp;by&nbsp;<\/p>\n<p style=\"position:absolute;top:189px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:211px;left:211px;white-space:nowrap\" class=\"ft50\">v<\/p>\n<p style=\"position:absolute;top:215px;left:221px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:211px;left:229px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;=&nbsp;<\/p>\n<p style=\"position:absolute;top:204px;left:272px;white-space:nowrap\" class=\"ft50\">&omega;\/k &nbsp;or &nbsp;&nbsp;&omega;&nbsp;= k.v<\/p>\n<p style=\"position:absolute;top:215px;left:458px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:210px;left:466px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:232px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:247px;left:157px;white-space:nowrap\" class=\"ft50\">The&nbsp;group velocity&nbsp;v<\/p>\n<p style=\"position:absolute;top:253px;left:303px;white-space:nowrap\" class=\"ft54\">g<\/p>\n<p style=\"position:absolute;top:247px;left:309px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;is given&nbsp;by&nbsp;<\/p>\n<p style=\"position:absolute;top:266px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:292px;left:211px;white-space:nowrap\" class=\"ft50\">v<\/p>\n<p style=\"position:absolute;top:296px;left:221px;white-space:nowrap\" class=\"ft54\">g<\/p>\n<p style=\"position:absolute;top:292px;left:229px;white-space:nowrap\" class=\"ft50\">&nbsp;=&nbsp;d<\/p>\n<p style=\"position:absolute;top:285px;left:272px;white-space:nowrap\" class=\"ft50\">&omega;&nbsp;=&nbsp;d(k.v<\/p>\n<p style=\"position:absolute;top:296px;left:370px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:292px;left:378px;white-space:nowrap\" class=\"ft50\">) dk&nbsp;<\/p>\n<p style=\"position:absolute;top:309px;left:265px;white-space:nowrap\" class=\"ft50\">dk<\/p>\n<p style=\"position:absolute;top:308px;left:286px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:333px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:352px;left:243px;white-space:nowrap\" class=\"ft50\">=<\/p>\n<p style=\"position:absolute;top:351px;left:254px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;v<\/p>\n<p style=\"position:absolute;top:356px;left:275px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:352px;left:282px;white-space:nowrap\" class=\"ft50\">&nbsp;+ k.dv<\/p>\n<p style=\"position:absolute;top:356px;left:358px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:352px;left:365px;white-space:nowrap\" class=\"ft50\">&nbsp;dk&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:369px;left:157px;white-space:nowrap\" class=\"ft511\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:406px;left:243px;white-space:nowrap\" class=\"ft50\">=<\/p>\n<p style=\"position:absolute;top:405px;left:254px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;v<\/p>\n<p style=\"position:absolute;top:410px;left:275px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:406px;left:283px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;+ (2<\/p>\n<p style=\"position:absolute;top:400px;left:347px;white-space:nowrap\" class=\"ft50\">&pi;\/&lambda;).&nbsp;dv<\/p>\n<p style=\"position:absolute;top:410px;left:432px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:406px;left:439px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:410px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:430px;left:383px;white-space:nowrap\" class=\"ft50\">d(2<\/p>\n<p style=\"position:absolute;top:423px;left:416px;white-space:nowrap\" class=\"ft50\">&pi;\/&lambda;)&nbsp;<\/p>\n<p style=\"position:absolute;top:451px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:481px;left:244px;white-space:nowrap\" class=\"ft50\">= v<\/p>\n<p style=\"position:absolute;top:485px;left:276px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:480px;left:283px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;+ (2&pi;\/&lambda;).(-&lambda;<\/p>\n<p style=\"position:absolute;top:470px;left:420px;white-space:nowrap\" class=\"ft55\">2<\/p>\n<p style=\"position:absolute;top:479px;left:429px;white-space:nowrap\" class=\"ft50\">\/2<\/p>\n<p style=\"position:absolute;top:472px;left:451px;white-space:nowrap\" class=\"ft50\">&pi;).dv<\/p>\n<p style=\"position:absolute;top:482px;left:504px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:478px;left:511px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:481px;left:601px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:501px;left:244px;white-space:nowrap\" class=\"ft56\">&nbsp;<\/p>\n<p style=\"position:absolute;top:501px;left:289px;white-space:nowrap\" class=\"ft56\">&nbsp;<\/p>\n<p style=\"position:absolute;top:501px;left:337px;white-space:nowrap\" class=\"ft56\">&nbsp;<\/p>\n<p style=\"position:absolute;top:499px;left:491px;white-space:nowrap\" class=\"ft50\">d<\/p>\n<p style=\"position:absolute;top:492px;left:502px;white-space:nowrap\" class=\"ft50\">&lambda;&nbsp;<\/p>\n<p style=\"position:absolute;top:501px;left:601px;white-space:nowrap\" class=\"ft56\">&nbsp;<\/p>\n<p style=\"position:absolute;top:541px;left:244px;white-space:nowrap\" class=\"ft50\">= v<\/p>\n<p style=\"position:absolute;top:545px;left:276px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:540px;left:283px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;&#8211;&nbsp;<\/p>\n<p style=\"position:absolute;top:535px;left:315px;white-space:nowrap\" class=\"ft50\">&lambda;. dv<\/p>\n<p style=\"position:absolute;top:545px;left:368px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:540px;left:375px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:541px;left:721px;white-space:nowrap\" class=\"ft50\">(6)<\/p>\n<p style=\"position:absolute;top:540px;left:754px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:558px;left:244px;white-space:nowrap\" class=\"ft57\">&nbsp;<\/p>\n<p style=\"position:absolute;top:558px;left:289px;white-space:nowrap\" class=\"ft57\">&nbsp;<\/p>\n<p style=\"position:absolute;top:558px;left:337px;white-space:nowrap\" class=\"ft58\">&nbsp;<\/p>\n<p style=\"position:absolute;top:558px;left:368px;white-space:nowrap\" class=\"ft57\">&nbsp;<\/p>\n<p style=\"position:absolute;top:558px;left:601px;white-space:nowrap\" class=\"ft57\">&nbsp;<\/p>\n<p style=\"position:absolute;top:565px;left:244px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:565px;left:289px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:565px;left:352px;white-space:nowrap\" class=\"ft50\">d<\/p>\n<p style=\"position:absolute;top:559px;left:362px;white-space:nowrap\" class=\"ft50\">&lambda;&nbsp;<\/p>\n<p style=\"position:absolute;top:565px;left:601px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:587px;left:157px;white-space:nowrap\" class=\"ft512\">&nbsp;<br \/>In the&nbsp;above&nbsp;expression,&nbsp;&nbsp;if&nbsp;&nbsp;(dv<\/p>\n<p style=\"position:absolute;top:615px;left:412px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:611px;left:420px;white-space:nowrap\" class=\"ft50\">\/d<\/p>\n<p style=\"position:absolute;top:604px;left:441px;white-space:nowrap\" class=\"ft50\">&lambda;) = 0,&nbsp;<\/p>\n<p style=\"position:absolute;top:604px;left:539px;white-space:nowrap\" class=\"ft50\">i.e., if&nbsp;the&nbsp;phase&nbsp;&nbsp;velocity does&nbsp;not&nbsp;<\/p>\n<p style=\"position:absolute;top:631px;left:157px;white-space:nowrap\" class=\"ft513\">depend on wavelength, then the&nbsp;group velocity&nbsp;and phase&nbsp;velocity&nbsp;are&nbsp;equal.&nbsp;Such a&nbsp;<br \/>medium is&nbsp;called&nbsp;a non-dispersive&nbsp;medium.&nbsp;In&nbsp;a dispersive&nbsp;medium,&nbsp;&nbsp;(dv<\/p>\n<p style=\"position:absolute;top:670px;left:746px;white-space:nowrap\" class=\"ft54\">p<\/p>\n<p style=\"position:absolute;top:666px;left:753px;white-space:nowrap\" class=\"ft50\">\/d<\/p>\n<p style=\"position:absolute;top:659px;left:775px;white-space:nowrap\" class=\"ft50\">&#61548;)&nbsp;&nbsp;is&nbsp;<\/p>\n<p style=\"position:absolute;top:686px;left:157px;white-space:nowrap\" class=\"ft514\">positive&nbsp;and&nbsp;hence&nbsp;the&nbsp;group velocity&nbsp;is&nbsp;less&nbsp;than the&nbsp;phase&nbsp;velocity.&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:738px;left:157px;white-space:nowrap\" class=\"ft51\"><b>Relation between &nbsp;group &nbsp;velocity&nbsp; and &nbsp;particle<\/b>&nbsp;<b>velocity&nbsp;(Velocity&nbsp;of&nbsp;de&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:764px;left:157px;white-space:nowrap\" class=\"ft51\"><b>Broglie waves):&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:790px;left:157px;white-space:nowrap\" class=\"ft515\">The&nbsp;phase&nbsp;velocity&nbsp;of waves&nbsp;depends&nbsp;on the&nbsp;wavelength.&nbsp;This&nbsp;is&nbsp;responsible&nbsp;for&nbsp;the&nbsp;well&nbsp;<br \/>known phenomenon&nbsp;of dispersion.&nbsp;In the&nbsp;case&nbsp;of light&nbsp;waves&nbsp;in vacuum, the&nbsp;phase&nbsp;<br \/>velocity&nbsp;is&nbsp;same&nbsp;for all wavelengths.&nbsp;<\/p>\n<p style=\"position:absolute;top:867px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:885px;left:157px;white-space:nowrap\" class=\"ft50\">In the&nbsp;case&nbsp;of de&nbsp;Broglie&nbsp;waves, we&nbsp;have,&nbsp;<\/p>\n<p style=\"position:absolute;top:885px;left:641px;white-space:nowrap\" class=\"ft510\">&nbsp;<\/p>\n<p style=\"position:absolute;top:922px;left:221px;white-space:nowrap\" class=\"ft50\">&omega;&nbsp;=&nbsp;2&pi;&nu;&nbsp;= 2&pi;mc<\/p>\n<p style=\"position:absolute;top:919px;left:369px;white-space:nowrap\" class=\"ft55\">2<\/p>\n<p style=\"position:absolute;top:928px;left:378px;white-space:nowrap\" class=\"ft50\">\/h =<\/p>\n<p style=\"position:absolute;top:927px;left:421px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:928px;left:454px;white-space:nowrap\" class=\"ft50\">2<\/p>\n<p style=\"position:absolute;top:922px;left:464px;white-space:nowrap\" class=\"ft50\">&pi;m<\/p>\n<p style=\"position:absolute;top:932px;left:485px;white-space:nowrap\" class=\"ft54\">0<\/p>\n<p style=\"position:absolute;top:928px;left:492px;white-space:nowrap\" class=\"ft50\">c<\/p>\n<p style=\"position:absolute;top:919px;left:503px;white-space:nowrap\" class=\"ft55\">2<\/p>\n<p style=\"position:absolute;top:927px;left:513px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:931px;left:715px;white-space:nowrap\" class=\"ft50\">(1)<\/p>\n<p style=\"position:absolute;top:930px;left:748px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:955px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:955px;left:427px;white-space:nowrap\" class=\"ft50\">h(1-v<\/p>\n<p style=\"position:absolute;top:946px;left:481px;white-space:nowrap\" class=\"ft55\">2<\/p>\n<p style=\"position:absolute;top:955px;left:490px;white-space:nowrap\" class=\"ft50\">\/c<\/p>\n<p style=\"position:absolute;top:946px;left:512px;white-space:nowrap\" class=\"ft55\">2<\/p>\n<p style=\"position:absolute;top:955px;left:521px;white-space:nowrap\" class=\"ft50\">)<\/p>\n<p style=\"position:absolute;top:946px;left:532px;white-space:nowrap\" class=\"ft55\">1\/2<\/p>\n<p style=\"position:absolute;top:954px;left:560px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:955px;left:641px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:978px;left:167px;white-space:nowrap\" class=\"ft50\">and&nbsp;&nbsp;k = 2<\/p>\n<p style=\"position:absolute;top:972px;left:262px;white-space:nowrap\" class=\"ft50\">&pi;\/&lambda;&nbsp;= 2&pi;mv\/h =&nbsp;<\/p>\n<p style=\"position:absolute;top:979px;left:487px;white-space:nowrap\" class=\"ft50\">2<\/p>\n<p style=\"position:absolute;top:972px;left:497px;white-space:nowrap\" class=\"ft50\">&pi;m<\/p>\n<p style=\"position:absolute;top:983px;left:518px;white-space:nowrap\" class=\"ft54\">0<\/p>\n<p style=\"position:absolute;top:979px;left:525px;white-space:nowrap\" class=\"ft50\">v<\/p>\n<p style=\"position:absolute;top:978px;left:536px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:979px;left:677px;white-space:nowrap\" class=\"ft510\">(2)<\/p>\n<p style=\"position:absolute;top:978px;left:708px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1000px;left:470px;white-space:nowrap\" class=\"ft50\">h(1-v<\/p>\n<p style=\"position:absolute;top:991px;left:524px;white-space:nowrap\" class=\"ft55\">2<\/p>\n<p style=\"position:absolute;top:1000px;left:534px;white-space:nowrap\" class=\"ft50\">\/c<\/p>\n<p style=\"position:absolute;top:991px;left:555px;white-space:nowrap\" class=\"ft55\">2<\/p>\n<p style=\"position:absolute;top:1000px;left:565px;white-space:nowrap\" class=\"ft50\">)<\/p>\n<p style=\"position:absolute;top:991px;left:575px;white-space:nowrap\" class=\"ft55\">1\/2<\/p>\n<p style=\"position:absolute;top:999px;left:604px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1014px;left:157px;white-space:nowrap\" class=\"ft50\">The&nbsp;group velocity&nbsp;of de&nbsp;Broglie&nbsp;waves&nbsp;is&nbsp;given by&nbsp;<\/p>\n<p style=\"position:absolute;top:1033px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1055px;left:190px;white-space:nowrap\" class=\"ft50\">V<\/p>\n<p style=\"position:absolute;top:1059px;left:200px;white-space:nowrap\" class=\"ft54\">g<\/p>\n<p style=\"position:absolute;top:1055px;left:208px;white-space:nowrap\" class=\"ft50\">&nbsp;&nbsp;= d<\/p>\n<p style=\"position:absolute;top:1048px;left:262px;white-space:nowrap\" class=\"ft50\">&omega;\/dk = &nbsp;d&omega;\/dv&nbsp;<\/p>\n<p style=\"position:absolute;top:1075px;left:340px;white-space:nowrap\" class=\"ft50\">dk\/dv<\/p>\n<p style=\"position:absolute;top:1074px;left:394px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1091px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1106px;left:157px;white-space:nowrap\" class=\"ft50\">&nbsp;<\/p>\n<\/div>\n<div id=\"page6-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224006.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft60\">5&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:57px;left:190px;white-space:nowrap\" class=\"ft60\">d<\/p>\n<p style=\"position:absolute;top:50px;left:200px;white-space:nowrap\" class=\"ft60\">&omega;\/dv&nbsp;=&nbsp;(2&pi;m<\/p>\n<p style=\"position:absolute;top:61px;left:316px;white-space:nowrap\" class=\"ft61\">0<\/p>\n<p style=\"position:absolute;top:57px;left:323px;white-space:nowrap\" class=\"ft60\">c<\/p>\n<p style=\"position:absolute;top:48px;left:333px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:57px;left:343px;white-space:nowrap\" class=\"ft60\">\/h).d(1-v<\/p>\n<p style=\"position:absolute;top:48px;left:437px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:57px;left:446px;white-space:nowrap\" class=\"ft60\">\/c<\/p>\n<p style=\"position:absolute;top:48px;left:467px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:57px;left:476px;white-space:nowrap\" class=\"ft60\">)<\/p>\n<p style=\"position:absolute;top:48px;left:487px;white-space:nowrap\" class=\"ft62\">1\/2<\/p>\n<p style=\"position:absolute;top:56px;left:514px;white-space:nowrap\" class=\"ft60\">&nbsp;&nbsp;=&nbsp;2<\/p>\n<p style=\"position:absolute;top:52px;left:556px;white-space:nowrap\" class=\"ft60\">&pi;m<\/p>\n<p style=\"position:absolute;top:63px;left:577px;white-space:nowrap\" class=\"ft61\">0<\/p>\n<p style=\"position:absolute;top:59px;left:584px;white-space:nowrap\" class=\"ft60\">v<\/p>\n<p style=\"position:absolute;top:58px;left:595px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:59px;left:703px;white-space:nowrap\" class=\"ft60\">(3)<\/p>\n<p style=\"position:absolute;top:58px;left:736px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:86px;left:383px;white-space:nowrap\" class=\"ft60\">dv<\/p>\n<p style=\"position:absolute;top:85px;left:405px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:84px;left:524px;white-space:nowrap\" class=\"ft60\">h(1-v<\/p>\n<p style=\"position:absolute;top:75px;left:575px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:84px;left:584px;white-space:nowrap\" class=\"ft60\">\/c<\/p>\n<p style=\"position:absolute;top:75px;left:605px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:84px;left:614px;white-space:nowrap\" class=\"ft60\">)<\/p>\n<p style=\"position:absolute;top:75px;left:624px;white-space:nowrap\" class=\"ft62\">3\/2<\/p>\n<p style=\"position:absolute;top:83px;left:651px;white-space:nowrap\" class=\"ft60\">&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:139px;left:190px;white-space:nowrap\" class=\"ft60\">dk\/dv = &nbsp;____2<\/p>\n<p style=\"position:absolute;top:133px;left:341px;white-space:nowrap\" class=\"ft60\">&pi;m<\/p>\n<p style=\"position:absolute;top:143px;left:362px;white-space:nowrap\" class=\"ft61\">0_____<\/p>\n<p style=\"position:absolute;top:138px;left:405px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:139px;left:517px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:139px;left:703px;white-space:nowrap\" class=\"ft60\">(4)<\/p>\n<p style=\"position:absolute;top:138px;left:736px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:167px;left:286px;white-space:nowrap\" class=\"ft60\">h(1-v<\/p>\n<p style=\"position:absolute;top:157px;left:340px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:167px;left:349px;white-space:nowrap\" class=\"ft60\">\/c<\/p>\n<p style=\"position:absolute;top:157px;left:371px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:167px;left:380px;white-space:nowrap\" class=\"ft60\">)<\/p>\n<p style=\"position:absolute;top:157px;left:391px;white-space:nowrap\" class=\"ft62\">3\/2<\/p>\n<p style=\"position:absolute;top:166px;left:419px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:169px;left:517px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:169px;left:655px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:190px;left:157px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:205px;left:167px;white-space:nowrap\" class=\"ft60\">From&nbsp;equations&nbsp;3&nbsp;and 4&nbsp;we&nbsp;get,&nbsp;<\/p>\n<p style=\"position:absolute;top:228px;left:157px;white-space:nowrap\" class=\"ft63\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:273px;left:221px;white-space:nowrap\" class=\"ft60\">v<\/p>\n<p style=\"position:absolute;top:277px;left:232px;white-space:nowrap\" class=\"ft61\">g<\/p>\n<p style=\"position:absolute;top:273px;left:239px;white-space:nowrap\" class=\"ft60\">&nbsp;&nbsp;= v<\/p>\n<p style=\"position:absolute;top:272px;left:293px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:285px;left:157px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:298px;left:157px;white-space:nowrap\" class=\"ft60\">Thus,&nbsp;the&nbsp;group velocity&nbsp;associated with de&nbsp;Broglie&nbsp;waves&nbsp;is&nbsp;just&nbsp;equal to&nbsp;the&nbsp;velocity&nbsp;<\/p>\n<p style=\"position:absolute;top:320px;left:157px;white-space:nowrap\" class=\"ft60\">with which the&nbsp;particle&nbsp;is&nbsp;moving.&nbsp;If&nbsp;we&nbsp;try&nbsp;to&nbsp;calculate&nbsp;the&nbsp;phase&nbsp;velocity,&nbsp;<\/p>\n<p style=\"position:absolute;top:335px;left:157px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:361px;left:190px;white-space:nowrap\" class=\"ft60\">V<\/p>\n<p style=\"position:absolute;top:365px;left:200px;white-space:nowrap\" class=\"ft61\">p<\/p>\n<p style=\"position:absolute;top:361px;left:208px;white-space:nowrap\" class=\"ft60\">=&nbsp;<\/p>\n<p style=\"position:absolute;top:354px;left:229px;white-space:nowrap\" class=\"ft60\">&omega;\/k&nbsp;= c<\/p>\n<p style=\"position:absolute;top:352px;left:306px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:361px;left:316px;white-space:nowrap\" class=\"ft60\">\/v = c<\/p>\n<p style=\"position:absolute;top:352px;left:381px;white-space:nowrap\" class=\"ft62\">2<\/p>\n<p style=\"position:absolute;top:361px;left:390px;white-space:nowrap\" class=\"ft60\">\/v<\/p>\n<p style=\"position:absolute;top:365px;left:412px;white-space:nowrap\" class=\"ft61\">g<\/p>\n<p style=\"position:absolute;top:360px;left:419px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:363px;left:724px;white-space:nowrap\" class=\"ft60\">(5)<\/p>\n<p style=\"position:absolute;top:362px;left:757px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:373px;left:157px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:386px;left:157px;white-space:nowrap\" class=\"ft65\">Since&nbsp;the&nbsp;group&nbsp;velocity&nbsp;or&nbsp;the&nbsp;particle&nbsp;velocity&nbsp;is&nbsp;always&nbsp;less&nbsp;than c, the&nbsp;phase&nbsp;velocity&nbsp;of&nbsp;<br \/>de&nbsp;Broglie&nbsp;waves&nbsp;turn out to&nbsp;be&nbsp;greater&nbsp;than c.&nbsp;&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:549px;left:165px;white-space:nowrap\" class=\"ft60\">S.NO&nbsp;<\/p>\n<p style=\"position:absolute;top:549px;left:331px;white-space:nowrap\" class=\"ft60\">RGPV QUESTIONS&nbsp;<\/p>\n<p style=\"position:absolute;top:549px;left:630px;white-space:nowrap\" class=\"ft60\">Year&nbsp;<\/p>\n<p style=\"position:absolute;top:549px;left:737px;white-space:nowrap\" class=\"ft60\">Marks&nbsp;<\/p>\n<p style=\"position:absolute;top:572px;left:165px;white-space:nowrap\" class=\"ft60\">Q.1&nbsp;&nbsp;Define&nbsp;group velocity&nbsp;and phase&nbsp;velocity.&nbsp;Prove&nbsp;<\/p>\n<p style=\"position:absolute;top:594px;left:218px;white-space:nowrap\" class=\"ft60\">that for&nbsp;a&nbsp;relativistic&nbsp;particle&nbsp;and non-&nbsp;relativistic&nbsp;<\/p>\n<p style=\"position:absolute;top:616px;left:218px;white-space:nowrap\" class=\"ft60\">particle,&nbsp;phase&nbsp;velocity&nbsp;is&nbsp;not&nbsp;equal to&nbsp;particle&nbsp;<\/p>\n<p style=\"position:absolute;top:638px;left:218px;white-space:nowrap\" class=\"ft60\">velocity.&nbsp;<\/p>\n<p style=\"position:absolute;top:572px;left:609px;white-space:nowrap\" class=\"ft60\">June&nbsp;2013&nbsp;<\/p>\n<p style=\"position:absolute;top:572px;left:755px;white-space:nowrap\" class=\"ft60\">7&nbsp;<\/p>\n<p style=\"position:absolute;top:660px;left:165px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:660px;left:218px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:660px;left:646px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:660px;left:760px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:683px;left:165px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:683px;left:218px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:683px;left:646px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:683px;left:760px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:706px;left:157px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:728px;left:157px;white-space:nowrap\" class=\"ft66\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:784px;left:459px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:805px;left:108px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<p style=\"position:absolute;top:805px;left:324px;white-space:nowrap\" class=\"ft60\">&nbsp;<\/p>\n<\/div>\n<div id=\"page7-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224007.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft70\">6&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:385px;white-space:nowrap\" class=\"ft71\"><b>Unit-01\/Lecture-03&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:85px;left:108px;white-space:nowrap\" class=\"ft72\"><b>&nbsp;&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:118px;left:108px;white-space:nowrap\" class=\"ft71\"><b>Heisenberg&nbsp;Uncertainty&nbsp;Principle[&nbsp;Rgpv&nbsp;Dec&nbsp;2012(7)]&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:155px;left:108px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:177px;left:162px;white-space:nowrap\" class=\"ft70\">This&nbsp;equation states&nbsp;that the&nbsp;product of&nbsp;uncertainty&nbsp;&#8710;x&nbsp;in the&nbsp;position&nbsp;of&nbsp;an&nbsp;object at some&nbsp;<\/p>\n<p style=\"position:absolute;top:207px;left:108px;white-space:nowrap\" class=\"ft76\">instant&nbsp;and&nbsp;the&nbsp;uncertainty&nbsp;in&nbsp;the&nbsp;momentum&nbsp;&#8710;p&nbsp;in&nbsp;the&nbsp;x-direction at the&nbsp;same&nbsp;instant is&nbsp;equal&nbsp;to&nbsp;<br \/>or greater than&nbsp;&#295;\/2.&nbsp;<\/p>\n<p style=\"position:absolute;top:298px;left:186px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:291px;left:197px;white-space:nowrap\" class=\"ft70\">&#8710;x.&nbsp;&#8710;p&nbsp;&ge;&nbsp;&nbsp;&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:297px;left:671px;white-space:nowrap\" class=\"ft70\">(1)&nbsp;<\/p>\n<p style=\"position:absolute;top:318px;left:108px;white-space:nowrap\" class=\"ft73\">&nbsp;<\/p>\n<p style=\"position:absolute;top:318px;left:303px;white-space:nowrap\" class=\"ft70\">2&nbsp;<\/p>\n<p style=\"position:absolute;top:318px;left:578px;white-space:nowrap\" class=\"ft73\">&nbsp;<\/p>\n<p style=\"position:absolute;top:334px;left:108px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:350px;left:108px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:364px;left:162px;white-space:nowrap\" class=\"ft70\">Another&nbsp;form&nbsp;of&nbsp;uncertainty&nbsp;principle&nbsp;relates&nbsp;energy&nbsp;and time.&nbsp;In the&nbsp;atomic&nbsp;process, if&nbsp;<\/p>\n<p style=\"position:absolute;top:390px;left:108px;white-space:nowrap\" class=\"ft70\">energy E is&nbsp;<\/p>\n<p style=\"position:absolute;top:392px;left:191px;white-space:nowrap\" class=\"ft74\">emitted&nbsp;&nbsp;as&nbsp;&nbsp;an&nbsp;&nbsp;electromagnetic&nbsp;&nbsp;wave&nbsp;&nbsp;during&nbsp;&nbsp;an&nbsp; interval&nbsp;<\/p>\n<p style=\"position:absolute;top:390px;left:591px;white-space:nowrap\" class=\"ft70\">of&nbsp;time&nbsp;&#8710;t, then, the&nbsp;uncertainty&nbsp;<\/p>\n<p style=\"position:absolute;top:417px;left:108px;white-space:nowrap\" class=\"ft78\">&#8710;E&nbsp;in&nbsp;the&nbsp;measured&nbsp;value&nbsp;of&nbsp;E&nbsp;depends&nbsp;on&nbsp;the&nbsp;duration&nbsp;of&nbsp;the&nbsp;time&nbsp;interval&nbsp;&#8710;t&nbsp;according&nbsp;to&nbsp;the&nbsp;<br \/>equation,&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:482px;left:162px;white-space:nowrap\" class=\"ft70\">&#8710;E.&nbsp;&#8710;t&nbsp;<\/p>\n<p style=\"position:absolute;top:482px;left:213px;white-space:nowrap\" class=\"ft70\">&ge; &#295;\/2&nbsp;<\/p>\n<p style=\"position:absolute;top:488px;left:633px;white-space:nowrap\" class=\"ft70\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<\/p>\n<p style=\"position:absolute;top:489px;left:682px;white-space:nowrap\" class=\"ft73\">(2)<\/p>\n<p style=\"position:absolute;top:488px;left:703px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:501px;left:108px;white-space:nowrap\" class=\"ft710\">&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:535px;left:113px;white-space:nowrap\" class=\"ft70\">Consider&nbsp;the&nbsp;combination of two&nbsp;waves&nbsp;represented by&nbsp;the&nbsp;formula&nbsp;<\/p>\n<p style=\"position:absolute;top:521px;left:769px;white-space:nowrap\" class=\"ft711\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:573px;left:108px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:573px;left:727px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:585px;left:162px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:607px;left:195px;white-space:nowrap\" class=\"ft70\">y<\/p>\n<p style=\"position:absolute;top:611px;left:206px;white-space:nowrap\" class=\"ft75\">1<\/p>\n<p style=\"position:absolute;top:607px;left:213px;white-space:nowrap\" class=\"ft70\">&nbsp;&nbsp;= A cos (<\/p>\n<p style=\"position:absolute;top:600px;left:332px;white-space:nowrap\" class=\"ft70\">&omega;t-kx)&nbsp;<\/p>\n<p style=\"position:absolute;top:626px;left:162px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:649px;left:194px;white-space:nowrap\" class=\"ft70\">y<\/p>\n<p style=\"position:absolute;top:652px;left:205px;white-space:nowrap\" class=\"ft75\">2<\/p>\n<p style=\"position:absolute;top:649px;left:212px;white-space:nowrap\" class=\"ft70\">&nbsp;= A cos {(<\/p>\n<p style=\"position:absolute;top:642px;left:331px;white-space:nowrap\" class=\"ft70\">&omega;+&#8710;&omega;)t&nbsp;&ndash;&nbsp;(k+&#8710;k)x}&nbsp;<\/p>\n<p style=\"position:absolute;top:667px;left:194px;white-space:nowrap\" class=\"ft70\">The&nbsp;resultant displacement is&nbsp;given by&nbsp;<\/p>\n<p style=\"position:absolute;top:687px;left:162px;white-space:nowrap\" class=\"ft712\">&nbsp;<br \/>y<\/p>\n<p style=\"position:absolute;top:704px;left:173px;white-space:nowrap\" class=\"ft70\">&nbsp;&nbsp;= y<\/p>\n<p style=\"position:absolute;top:709px;left:215px;white-space:nowrap\" class=\"ft75\">1<\/p>\n<p style=\"position:absolute;top:705px;left:223px;white-space:nowrap\" class=\"ft70\">&nbsp;&nbsp;+ y<\/p>\n<p style=\"position:absolute;top:709px;left:277px;white-space:nowrap\" class=\"ft75\">2<\/p>\n<p style=\"position:absolute;top:705px;left:284px;white-space:nowrap\" class=\"ft70\">&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:729px;left:135px;white-space:nowrap\" class=\"ft70\">&nbsp; &nbsp; &nbsp;= 2A cos {(<\/p>\n<p style=\"position:absolute;top:722px;left:308px;white-space:nowrap\" class=\"ft70\">&omega;+&omega;+&#8710;&omega;)t&ndash;(k+k+&#8710;k)x} cos&nbsp;(&#8710;&omega;t-&#8710;kx)&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:749px;left:333px;white-space:nowrap\" class=\"ft70\">2<\/p>\n<p style=\"position:absolute;top:748px;left:344px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:749px;left:549px;white-space:nowrap\" class=\"ft70\">2<\/p>\n<p style=\"position:absolute;top:748px;left:560px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:765px;left:122px;white-space:nowrap\" class=\"ft70\">&nbsp;&nbsp;&asymp;&nbsp;2A&nbsp;cos(&omega;t&ndash;kx).cos(&#8710;&omega;t\/2-&#8710;kx\/2)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:793px;left:476px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:793px;left:662px;white-space:nowrap\" class=\"ft70\">(3)<\/p>\n<p style=\"position:absolute;top:792px;left:695px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:808px;left:108px;white-space:nowrap\" class=\"ft70\">The&nbsp;&nbsp;velocity &nbsp;of&nbsp; the &nbsp;resultant&nbsp; wave &nbsp;is&nbsp; given&nbsp; by the speed&nbsp;with which a&nbsp;reference&nbsp;point, say&nbsp;<\/p>\n<p style=\"position:absolute;top:830px;left:108px;white-space:nowrap\" class=\"ft70\">the&nbsp;maximum&nbsp;amplitude&nbsp;point, moves.&nbsp;Taking&nbsp;the&nbsp;amplitude&nbsp;of the&nbsp;resultant wave&nbsp;as&nbsp;constant,&nbsp;<\/p>\n<p style=\"position:absolute;top:860px;left:151px;white-space:nowrap\" class=\"ft70\">2A cos(<\/p>\n<p style=\"position:absolute;top:853px;left:227px;white-space:nowrap\" class=\"ft70\">&#8710;&omega;t\/2-&#8710;kx\/2) =&nbsp;constant&nbsp;<\/p>\n<p style=\"position:absolute;top:881px;left:108px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:905px;left:165px;white-space:nowrap\" class=\"ft70\">or<\/p>\n<p style=\"position:absolute;top:904px;left:187px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:905px;left:222px;white-space:nowrap\" class=\"ft70\">(<\/p>\n<p style=\"position:absolute;top:898px;left:233px;white-space:nowrap\" class=\"ft70\">&#8710;&omega;t\/2-&#8710;kx\/2) =&nbsp;constant&nbsp;<\/p>\n<p style=\"position:absolute;top:921px;left:108px;white-space:nowrap\" class=\"ft70\">for &nbsp;maximum amplitude&nbsp;&nbsp;cos(&#8710;&omega;t\/2-&#8710;kx\/2)=0&nbsp;<\/p>\n<p style=\"position:absolute;top:943px;left:108px;white-space:nowrap\" class=\"ft70\">Thus,&nbsp;<\/p>\n<p style=\"position:absolute;top:974px;left:165px;white-space:nowrap\" class=\"ft70\">(<\/p>\n<p style=\"position:absolute;top:967px;left:176px;white-space:nowrap\" class=\"ft70\">&#8710;&omega;t\/2-&#8710;<\/p>\n<p style=\"position:absolute;top:967px;left:253px;white-space:nowrap\" class=\"ft70\">kx\/2)= n&pi;\/2.&nbsp;<\/p>\n<p style=\"position:absolute;top:994px;left:165px;white-space:nowrap\" class=\"ft70\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1009px;left:108px;white-space:nowrap\" class=\"ft70\">Let the&nbsp;displacement of two&nbsp;successive&nbsp;nodes&nbsp;be&nbsp;x1&nbsp;and x2, then&nbsp;<\/p>\n<p style=\"position:absolute;top:1040px;left:165px;white-space:nowrap\" class=\"ft70\">&nbsp;(<\/p>\n<p style=\"position:absolute;top:1033px;left:187px;white-space:nowrap\" class=\"ft70\">&#8710;&omega;t\/2-&#8710;kx1\/2)=&nbsp;<\/p>\n<p style=\"position:absolute;top:1033px;left:351px;white-space:nowrap\" class=\"ft70\">&pi;\/2&nbsp;<\/p>\n<p style=\"position:absolute;top:1063px;left:165px;white-space:nowrap\" class=\"ft70\">(<\/p>\n<p style=\"position:absolute;top:1056px;left:176px;white-space:nowrap\" class=\"ft70\">&#8710;&omega;t\/2-&#8710;kx2\/2)=&nbsp;<\/p>\n<p style=\"position:absolute;top:1056px;left:340px;white-space:nowrap\" class=\"ft70\">3&pi;\/2&nbsp;<\/p>\n<p style=\"position:absolute;top:1083px;left:165px;white-space:nowrap\" class=\"ft713\">On solving, we get&nbsp;<br \/>&#8710;k(x2-<\/p>\n<p style=\"position:absolute;top:1100px;left:230px;white-space:nowrap\" class=\"ft70\">x1)= &pi;&nbsp;<\/p>\n<\/div>\n<div id=\"page8-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224008.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft80\">7&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:48px;left:165px;white-space:nowrap\" class=\"ft80\">&#8710;&nbsp;x=&nbsp;<\/p>\n<p style=\"position:absolute;top:48px;left:219px;white-space:nowrap\" class=\"ft80\">&pi;\/&#8710;k, where&nbsp;&#8710;k=2&nbsp;&pi;\/&lambda;&nbsp;<\/p>\n<p style=\"position:absolute;top:71px;left:108px;white-space:nowrap\" class=\"ft81\"><b>&nbsp;<\/b>&#8710;&nbsp;x=&lambda;\/2&nbsp;<\/p>\n<p style=\"position:absolute;top:110px;left:108px;white-space:nowrap\" class=\"ft80\">=h\/2<\/p>\n<p style=\"position:absolute;top:104px;left:141px;white-space:nowrap\" class=\"ft80\">&#8710;p&nbsp;<\/p>\n<p style=\"position:absolute;top:137px;left:108px;white-space:nowrap\" class=\"ft80\">&#8710;&nbsp;x.&nbsp;&#8710;p=&nbsp;h\/2&nbsp;<\/p>\n<p style=\"position:absolute;top:176px;left:108px;white-space:nowrap\" class=\"ft80\">Or&nbsp;<\/p>\n<p style=\"position:absolute;top:170px;left:131px;white-space:nowrap\" class=\"ft80\">&#8710;&nbsp;x.&nbsp;&#8710;<\/p>\n<p style=\"position:absolute;top:170px;left:190px;white-space:nowrap\" class=\"ft80\">p&asymp;&nbsp;h&nbsp;<\/p>\n<p style=\"position:absolute;top:202px;left:108px;white-space:nowrap\" class=\"ft82\">This&nbsp;is&nbsp;the&nbsp;required principle.&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:256px;left:108px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:278px;left:108px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:300px;left:108px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:322px;left:108px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:344px;left:116px;white-space:nowrap\" class=\"ft80\">S.NO&nbsp;<\/p>\n<p style=\"position:absolute;top:344px;left:320px;white-space:nowrap\" class=\"ft80\">RGPV&nbsp;QUESTIONS&nbsp;<\/p>\n<p style=\"position:absolute;top:344px;left:646px;white-space:nowrap\" class=\"ft80\">Year&nbsp;<\/p>\n<p style=\"position:absolute;top:344px;left:752px;white-space:nowrap\" class=\"ft80\">Marks&nbsp;<\/p>\n<p style=\"position:absolute;top:367px;left:116px;white-space:nowrap\" class=\"ft80\">Q.1&nbsp;<\/p>\n<p style=\"position:absolute;top:367px;left:177px;white-space:nowrap\" class=\"ft80\">Explain the&nbsp;concept of&nbsp;wave&nbsp;packet&nbsp;and give&nbsp;the&nbsp;<\/p>\n<p style=\"position:absolute;top:389px;left:177px;white-space:nowrap\" class=\"ft80\">mathematical&nbsp;proof of Heisenberg&rsquo;s&nbsp;uncertainty&nbsp;<\/p>\n<p style=\"position:absolute;top:411px;left:177px;white-space:nowrap\" class=\"ft80\">principle?&nbsp;<\/p>\n<p style=\"position:absolute;top:367px;left:609px;white-space:nowrap\" class=\"ft80\">Dec&nbsp;2012&nbsp;<\/p>\n<p style=\"position:absolute;top:367px;left:730px;white-space:nowrap\" class=\"ft80\">7&nbsp;<\/p>\n<p style=\"position:absolute;top:434px;left:108px;white-space:nowrap\" class=\"ft83\">&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:697px;left:468px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:720px;left:459px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:742px;left:108px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:764px;left:108px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<p style=\"position:absolute;top:764px;left:324px;white-space:nowrap\" class=\"ft80\">&nbsp;<\/p>\n<\/div>\n<div id=\"page9-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224009.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft90\">8&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft94\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:68px;left:376px;white-space:nowrap\" class=\"ft91\"><b>Unit-01\/Lecture-04&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:107px;left:108px;white-space:nowrap\" class=\"ft95\"><b>&nbsp;&nbsp;<br \/>Applications&nbsp;of&nbsp;uncertainty&nbsp;principle:&nbsp;[Rgpv&nbsp;June&nbsp;2013(7)]&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:162px;left:108px;white-space:nowrap\" class=\"ft90\">(a).&nbsp;The&nbsp;uncertainty&nbsp;principle&nbsp;has&nbsp;far&nbsp;reaching&nbsp;implications.&nbsp;In&nbsp;fact, it&nbsp;has&nbsp;been very&nbsp;useful&nbsp;in&nbsp;<\/p>\n<p style=\"position:absolute;top:184px;left:108px;white-space:nowrap\" class=\"ft90\">explaining&nbsp;many&nbsp;observations&nbsp;which cannot be&nbsp;explained otherwise.&nbsp;A&nbsp;few of the&nbsp;applications&nbsp;of&nbsp;<\/p>\n<p style=\"position:absolute;top:206px;left:108px;white-space:nowrap\" class=\"ft90\">the&nbsp;uncertainty&nbsp;principle&nbsp;are&nbsp;worth mentioning&nbsp;<\/p>\n<p style=\"position:absolute;top:228px;left:108px;white-space:nowrap\" class=\"ft90\">&nbsp;<\/p>\n<p style=\"position:absolute;top:250px;left:108px;white-space:nowrap\" class=\"ft98\">We&nbsp;have&nbsp;the&nbsp;following&nbsp;&lsquo;Thought experiment&rsquo; to&nbsp;illustrate&nbsp;the&nbsp;uncertainty&nbsp;principle.&nbsp;Imagine&nbsp;an&nbsp;<br \/>electron being&nbsp;observed using&nbsp;a&nbsp;microscope.&nbsp;&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>The&nbsp;process&nbsp;of&nbsp;observation&nbsp;involves&nbsp;a&nbsp;photon&nbsp;of&nbsp;wavelength&nbsp;&lambda;&nbsp;incident on&nbsp;the&nbsp;electron and&nbsp;<br \/>getting&nbsp;scattered into&nbsp;the&nbsp;microscope.&nbsp;The&nbsp;event may&nbsp;be&nbsp;considered as&nbsp;a&nbsp;two-body&nbsp;problem&nbsp;in&nbsp;<br \/>which a&nbsp;photon interacts&nbsp;with an electron.&nbsp;The&nbsp;change&nbsp;in the&nbsp;velocity&nbsp;of&nbsp;the&nbsp;photon&nbsp;during&nbsp;the&nbsp;<br \/>interaction may&nbsp;be&nbsp;anything&nbsp;between zero&nbsp;(for&nbsp;grazing&nbsp;angle&nbsp;of incidence)&nbsp;and 2c&nbsp;(for&nbsp;head-on&nbsp;<br \/>collision&nbsp;and&nbsp;reflection).&nbsp;The&nbsp;average&nbsp;change&nbsp;in the&nbsp;momentum&nbsp;of&nbsp;the&nbsp;photon&nbsp;may&nbsp;be&nbsp;written as&nbsp;<br \/>equal&nbsp;to&nbsp;(h&nu;\/c)&nbsp;or (h\/&lambda;).This&nbsp;difference&nbsp;in&nbsp;momentum&nbsp;is&nbsp;carried&nbsp;by&nbsp;the&nbsp;recoiling&nbsp;electron&nbsp;which&nbsp;<br \/>was&nbsp;initially&nbsp;at rest.&nbsp;The&nbsp;change&nbsp;or&nbsp;uncertainty&nbsp;in the&nbsp;momentum&nbsp;of the&nbsp;electron may&nbsp;thus&nbsp;be&nbsp;<br \/>written&nbsp;as&nbsp;(h\/&lambda;).&nbsp;&nbsp;At the&nbsp;same&nbsp;time, the&nbsp;position of the&nbsp;electron&nbsp;can be&nbsp;<\/p>\n<p style=\"position:absolute;top:890px;left:681px;white-space:nowrap\" class=\"ft93\">determined to&nbsp;an&nbsp;<\/p>\n<p style=\"position:absolute;top:923px;left:108px;white-space:nowrap\" class=\"ft93\">accuracy limited&nbsp;by the&nbsp;resolving&nbsp;<\/p>\n<p style=\"position:absolute;top:923px;left:346px;white-space:nowrap\" class=\"ft90\">power&nbsp;of the&nbsp;microscope,&nbsp;which is&nbsp;of&nbsp;the&nbsp;order&nbsp;of&nbsp;&lambda;.&nbsp;Hence,&nbsp;<\/p>\n<p style=\"position:absolute;top:955px;left:108px;white-space:nowrap\" class=\"ft96\">the&nbsp;product&nbsp;of the&nbsp;uncertainties&nbsp;in position and&nbsp;momentum&nbsp;is&nbsp;of the&nbsp;order&nbsp;of h.&nbsp;This&nbsp;argument&nbsp;<br \/>implies&nbsp;that the&nbsp;uncertainty&nbsp;is&nbsp;associated with the&nbsp;measuring&nbsp;process.&nbsp;The&nbsp;illustration only&nbsp;<br \/>estimates&nbsp;the accuracy&nbsp;of&nbsp;measurement,&nbsp;the uncertainty&nbsp;being&nbsp;inherent in the&nbsp;nature&nbsp;of the&nbsp;<br \/>moving&nbsp;particles&nbsp;involved.&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<\/div>\n<div id=\"page10-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224010.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:455px;white-space:nowrap\" class=\"ft100\">9&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft108\">&nbsp;<br \/>&nbsp;<br \/><b>b).&nbsp;Diffraction&nbsp;of&nbsp;a&nbsp;beam of&nbsp;electrons<\/b>:&nbsp;Diffraction&nbsp;of<b>&nbsp;<\/b>a&nbsp;beam of&nbsp;electrons&nbsp;at&nbsp;a slit&nbsp;is&nbsp;the&nbsp;effect&nbsp;of&nbsp;<br \/>uncertainty&nbsp;principle.&nbsp;As&nbsp;the&nbsp;slit&nbsp;is&nbsp;made&nbsp;narrower,&nbsp;<br \/>Thereby&nbsp;reducing&nbsp;the&nbsp;uncertainty&nbsp;&nbsp;in the&nbsp;position of&nbsp;the&nbsp;electrons&nbsp;in the&nbsp;beam,&nbsp;the&nbsp;beam&nbsp;<br \/>spreads&nbsp;even&nbsp;more&nbsp;indicating&nbsp;a larger uncertainty&nbsp;in&nbsp;its&nbsp;velocity&nbsp;or momentum.&nbsp;<\/p>\n<p style=\"position:absolute;top:205px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:234px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:263px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:292px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:321px;left:108px;white-space:nowrap\" class=\"ft109\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:366px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:388px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:410px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:432px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:454px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:476px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:498px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:520px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:542px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:564px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:586px;left:119px;white-space:nowrap\" class=\"ft100\">Figure&nbsp;(2)&nbsp;shows&nbsp;the&nbsp;diffraction&nbsp;of&nbsp;an&nbsp;electron&nbsp;beam&nbsp;by&nbsp;a&nbsp;narrow&nbsp;slit&nbsp;of&nbsp;width&nbsp;&#8710;x.&nbsp;The&nbsp;beam&nbsp;<\/p>\n<p style=\"position:absolute;top:618px;left:108px;white-space:nowrap\" class=\"ft107\">travelling&nbsp;along&nbsp;OX&nbsp;is&nbsp;diffracted along&nbsp;OY&nbsp;through an angle&nbsp;&theta;.&nbsp;Due&nbsp;to&nbsp;the&nbsp;wave&nbsp;nature&nbsp;of the&nbsp;<br \/>electron, we&nbsp;observe&nbsp;Fraunhoffer&nbsp;diffraction on the&nbsp;screen placed along&nbsp;XY.&nbsp;The&nbsp;accuracy&nbsp;with&nbsp;<br \/>which the&nbsp;position&nbsp;of&nbsp;the&nbsp;electron is&nbsp;known&nbsp;is&nbsp;&#8710;x&nbsp;since&nbsp;it is&nbsp;uncertain from&nbsp;which&nbsp;place&nbsp;in&nbsp;the&nbsp;slit&nbsp;<br \/>the&nbsp;electron&nbsp;passes.&nbsp;According&nbsp;to&nbsp;the&nbsp;theory&nbsp;of diffraction, we&nbsp;have&nbsp;<\/p>\n<p style=\"position:absolute;top:736px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:749px;left:195px;white-space:nowrap\" class=\"ft100\">&lambda;&nbsp;=&nbsp;&#8710;x.sin&nbsp;&theta;&nbsp;<\/p>\n<p style=\"position:absolute;top:749px;left:333px;white-space:nowrap\" class=\"ft100\">or &#8710;x =&nbsp;&lambda;\/ sin&nbsp;&theta;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:772px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:793px;left:108px;white-space:nowrap\" class=\"ft100\">Further, the&nbsp;initial&nbsp;momentum&nbsp;of the&nbsp;electron along&nbsp;XY&nbsp;was&nbsp;zero&nbsp;and after&nbsp;diffraction, the&nbsp;<\/p>\n<p style=\"position:absolute;top:815px;left:108px;white-space:nowrap\" class=\"ft100\">momentum&nbsp;of the&nbsp;electron is&nbsp;p.&nbsp;sin&nbsp;&nbsp;&theta;&nbsp;&nbsp;where&nbsp;p is&nbsp;the&nbsp;&nbsp;momentum&nbsp;&nbsp;of the&nbsp;electron&nbsp;along&nbsp;the&nbsp;<\/p>\n<p style=\"position:absolute;top:837px;left:108px;white-space:nowrap\" class=\"ft1010\">incidence&nbsp;direction.&nbsp;Hence, the&nbsp;change&nbsp;in momentum&nbsp;of the&nbsp;electron along&nbsp;XY&nbsp;is&nbsp;p.&nbsp;sin&nbsp;&nbsp;&theta;&nbsp;&nbsp;or&nbsp;<br \/>(h\/<\/p>\n<p style=\"position:absolute;top:861px;left:134px;white-space:nowrap\" class=\"ft100\">&lambda;).&nbsp;Sin&theta;.&nbsp;<\/p>\n<p style=\"position:absolute;top:860px;left:226px;white-space:nowrap\" class=\"ft100\">Assuming&nbsp;the&nbsp;change&nbsp;in the&nbsp;momentum&nbsp;as&nbsp;representative&nbsp;of the&nbsp;uncertainty&nbsp;in&nbsp;<\/p>\n<p style=\"position:absolute;top:883px;left:108px;white-space:nowrap\" class=\"ft100\">momentum,&nbsp;we&nbsp;get&nbsp;<\/p>\n<p style=\"position:absolute;top:894px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:907px;left:227px;white-space:nowrap\" class=\"ft100\">&#8710;x.&nbsp;&#8710;p<\/p>\n<p style=\"position:absolute;top:918px;left:292px;white-space:nowrap\" class=\"ft102\">x<\/p>\n<p style=\"position:absolute;top:914px;left:299px;white-space:nowrap\" class=\"ft100\">&nbsp;&nbsp;= &nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:907px;left:364px;white-space:nowrap\" class=\"ft100\">&lambda;&nbsp;&nbsp;.h.sin&nbsp;&theta;&nbsp;=&nbsp;h&nbsp;<\/p>\n<p style=\"position:absolute;top:930px;left:227px;white-space:nowrap\" class=\"ft103\">&nbsp;<\/p>\n<p style=\"position:absolute;top:930px;left:331px;white-space:nowrap\" class=\"ft104\">&nbsp;<\/p>\n<p style=\"position:absolute;top:930px;left:385px;white-space:nowrap\" class=\"ft103\">&nbsp;<\/p>\n<p style=\"position:absolute;top:930px;left:392px;white-space:nowrap\" class=\"ft103\">&nbsp;<\/p>\n<p style=\"position:absolute;top:937px;left:227px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:938px;left:334px;white-space:nowrap\" class=\"ft100\">sin&nbsp;<\/p>\n<p style=\"position:absolute;top:931px;left:378px;white-space:nowrap\" class=\"ft100\">&theta;&nbsp;&nbsp;&lambda;&nbsp;<\/p>\n<p style=\"position:absolute;top:961px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:977px;left:108px;white-space:nowrap\" class=\"ft107\"><b>(c). Electron cannot reside&nbsp;in nucleus<\/b>:&nbsp;In beta&nbsp;decay,&nbsp;electrons&nbsp;are<b>&nbsp;<\/b>emitted from&nbsp;the&nbsp;nucleus&nbsp;of&nbsp;<br \/>the&nbsp;radioactive&nbsp;element.&nbsp;Assuming&nbsp;the&nbsp;diameter&nbsp;of the&nbsp;nucleus&nbsp;to&nbsp;represent the&nbsp;uncertainty&nbsp;in&nbsp;<br \/>the&nbsp;position&nbsp;of&nbsp;electron&nbsp;inside&nbsp;the&nbsp;nucleus, the&nbsp;uncertainty&nbsp;in the&nbsp;momentum&nbsp;can&nbsp;be&nbsp;calculated&nbsp;<br \/>as&nbsp;follows:&nbsp;<\/p>\n<p style=\"position:absolute;top:1091px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1114px;left:237px;white-space:nowrap\" class=\"ft100\">Radius&nbsp;of the&nbsp;nucleus&nbsp;=&nbsp;r&nbsp;=&nbsp;5 x 10<\/p>\n<p style=\"position:absolute;top:1110px;left:483px;white-space:nowrap\" class=\"ft105\">-15<\/p>\n<p style=\"position:absolute;top:1120px;left:504px;white-space:nowrap\" class=\"ft100\">&nbsp;m&nbsp;<\/p>\n<p style=\"position:absolute;top:1130px;left:108px;white-space:nowrap\" class=\"ft100\">&nbsp;<\/p>\n<\/div>\n<div id=\"page11-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224011.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft110\">10&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:49px;left:237px;white-space:nowrap\" class=\"ft110\">&#8710;x = 2r = 10<\/p>\n<p style=\"position:absolute;top:47px;left:367px;white-space:nowrap\" class=\"ft111\">-14<\/p>\n<p style=\"position:absolute;top:56px;left:395px;white-space:nowrap\" class=\"ft110\">&nbsp;m.<\/p>\n<p style=\"position:absolute;top:55px;left:427px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:70px;left:108px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:86px;left:112px;white-space:nowrap\" class=\"ft112\">&#8710;p = h\/2&pi;&#8710;x =&nbsp;6.62&#215;10<\/p>\n<p style=\"position:absolute;top:84px;left:329px;white-space:nowrap\" class=\"ft111\">-34<\/p>\n<p style=\"position:absolute;top:92px;left:356px;white-space:nowrap\" class=\"ft112\">\/(2&#215;3.14&#215;10<\/p>\n<p style=\"position:absolute;top:84px;left:470px;white-space:nowrap\" class=\"ft111\">-14<\/p>\n<p style=\"position:absolute;top:92px;left:497px;white-space:nowrap\" class=\"ft112\">) =&nbsp;<\/p>\n<p style=\"position:absolute;top:123px;left:241px;white-space:nowrap\" class=\"ft112\">1.055&#215;10<\/p>\n<p style=\"position:absolute;top:114px;left:324px;white-space:nowrap\" class=\"ft111\">-20<\/p>\n<p style=\"position:absolute;top:123px;left:351px;white-space:nowrap\" class=\"ft112\">&nbsp;kg m s<\/p>\n<p style=\"position:absolute;top:114px;left:424px;white-space:nowrap\" class=\"ft111\">-1<\/p>\n<p style=\"position:absolute;top:121px;left:442px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:134px;left:108px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:151px;left:184px;white-space:nowrap\" class=\"ft113\">&nbsp;<br \/>Assuming&nbsp;that the&nbsp;electron was&nbsp;at rest before&nbsp;its&nbsp;emission,&nbsp;the&nbsp;change&nbsp;in&nbsp;momentum&nbsp;<\/p>\n<p style=\"position:absolute;top:206px;left:108px;white-space:nowrap\" class=\"ft114\">can&nbsp;be&nbsp;taken&nbsp;as&nbsp;equal to&nbsp;its&nbsp;momentum.&nbsp;This&nbsp;magnitude&nbsp;of&nbsp;change&nbsp;in&nbsp;momentum indicates&nbsp;large&nbsp;<br \/>velocity for&nbsp;the electron.&nbsp;Hence,&nbsp;the&nbsp;energy of&nbsp;the emitted&nbsp;electron&nbsp;will&nbsp;be&nbsp;<\/p>\n<p style=\"position:absolute;top:262px;left:108px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:284px;left:155px;white-space:nowrap\" class=\"ft110\">E<\/p>\n<p style=\"position:absolute;top:282px;left:166px;white-space:nowrap\" class=\"ft110\">&nbsp;&nbsp;= pc = 1.055&#215;10<\/p>\n<p style=\"position:absolute;top:274px;left:339px;white-space:nowrap\" class=\"ft111\">-20<\/p>\n<p style=\"position:absolute;top:284px;left:367px;white-space:nowrap\" class=\"ft110\">&nbsp;&nbsp;x 3&#215;10<\/p>\n<p style=\"position:absolute;top:274px;left:454px;white-space:nowrap\" class=\"ft111\">8<\/p>\n<p style=\"position:absolute;top:284px;left:463px;white-space:nowrap\" class=\"ft110\">&nbsp;&nbsp;= 3.165 x 10<\/p>\n<p style=\"position:absolute;top:274px;left:614px;white-space:nowrap\" class=\"ft111\">-12<\/p>\n<p style=\"position:absolute;top:284px;left:643px;white-space:nowrap\" class=\"ft110\">&nbsp;&nbsp;J&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:291px;left:108px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:308px;left:230px;white-space:nowrap\" class=\"ft110\">=&nbsp;&nbsp;19.8 MeV.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:338px;left:108px;white-space:nowrap\" class=\"ft116\">&nbsp;<br \/>This&nbsp;indicates&nbsp;that the&nbsp;electrons&nbsp;inside&nbsp;the&nbsp;nucleus&nbsp;must have&nbsp;kinetic&nbsp;energy&nbsp;of 19.8&nbsp;MeV.&nbsp;But&nbsp;<br \/>the&nbsp;electrons&nbsp;emitted&nbsp;during&nbsp;beta&nbsp;decay&nbsp;have&nbsp;kinetic&nbsp;energy&nbsp;of the&nbsp;order&nbsp;of&nbsp;1&nbsp;MeV.&nbsp;This&nbsp;<br \/>indicates&nbsp;that electrons&nbsp;do&nbsp;not exist in the&nbsp;nucleus&nbsp;of the&nbsp;atom&nbsp;but are&nbsp;&lsquo;manufactured&rsquo; by&nbsp;the&nbsp;<br \/>nucleus&nbsp;at the&nbsp;time&nbsp;of decay.&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:536px;left:108px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:573px;left:108px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:588px;left:108px;white-space:nowrap\" class=\"ft117\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:644px;left:459px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:667px;left:108px;white-space:nowrap\" class=\"ft110\">S.NO&nbsp;<\/p>\n<p style=\"position:absolute;top:667px;left:311px;white-space:nowrap\" class=\"ft110\">RGPV QUESTIONS&nbsp;<\/p>\n<p style=\"position:absolute;top:667px;left:636px;white-space:nowrap\" class=\"ft110\">Year&nbsp;<\/p>\n<p style=\"position:absolute;top:667px;left:743px;white-space:nowrap\" class=\"ft110\">Marks&nbsp;<\/p>\n<p style=\"position:absolute;top:689px;left:108px;white-space:nowrap\" class=\"ft110\">Q.1&nbsp;<\/p>\n<p style=\"position:absolute;top:689px;left:168px;white-space:nowrap\" class=\"ft110\">State&nbsp;Heisenberg&rsquo;s&nbsp;uncertainty&nbsp;principle&nbsp;and derive&nbsp;it&nbsp;<\/p>\n<p style=\"position:absolute;top:711px;left:168px;white-space:nowrap\" class=\"ft110\">from hypothetical gamma ray&nbsp;microscope?&nbsp;<\/p>\n<p style=\"position:absolute;top:689px;left:599px;white-space:nowrap\" class=\"ft110\">June&nbsp;2013&nbsp;<\/p>\n<p style=\"position:absolute;top:689px;left:720px;white-space:nowrap\" class=\"ft110\">7&nbsp;<\/p>\n<p style=\"position:absolute;top:734px;left:108px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<p style=\"position:absolute;top:734px;left:324px;white-space:nowrap\" class=\"ft110\">&nbsp;<\/p>\n<\/div>\n<div id=\"page12-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224012.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft120\">11&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft120\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:376px;white-space:nowrap\" class=\"ft121\"><b>Unit-01\/Lecture-05&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:85px;left:108px;white-space:nowrap\" class=\"ft122\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:107px;left:108px;white-space:nowrap\" class=\"ft121\"><b>Compton Effect&nbsp;{Rgpv&nbsp;June&nbsp;2012(7),&nbsp;Dec&nbsp;2013&nbsp;(7)]&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:145px;left:108px;white-space:nowrap\" class=\"ft129\">He discovered&nbsp;that&nbsp;&ldquo;when&nbsp;a&nbsp;beam of&nbsp;monochromatic&nbsp;radiation&nbsp;(X-ray)&nbsp;of sharply&nbsp;defined&nbsp;frequency&nbsp;were&nbsp;<br \/>incident&nbsp;on&nbsp;a&nbsp;material&nbsp;of&nbsp;low&nbsp;atomic&nbsp;number&nbsp;(like&nbsp;carbon),&nbsp;the ray&nbsp;suffered&nbsp;a&nbsp;change&nbsp;of&nbsp;frequency&nbsp;on&nbsp;<br \/>scattering&rdquo;.&nbsp;The scattered&nbsp;beam&nbsp;contains&nbsp;two&nbsp;wavelengths. In&nbsp;addition&nbsp;to&nbsp;the&nbsp;expected&nbsp;incident&nbsp;<br \/>wavelength,&nbsp;there&nbsp;exists&nbsp;a&nbsp;line&nbsp;of&nbsp;longer&nbsp;wavelength.&nbsp;The&nbsp;change&nbsp;of&nbsp;wavelength is&nbsp;due&nbsp;to&nbsp;the&nbsp;loss&nbsp;of&nbsp;<br \/>energy&nbsp;of&nbsp;the&nbsp;incident&nbsp;rays. This&nbsp;phenomenon&nbsp;is&nbsp;known&nbsp;as&nbsp;<\/p>\n<p style=\"position:absolute;top:261px;left:499px;white-space:nowrap\" class=\"ft124\"><b>Compton Effect<\/b>.&nbsp;<\/p>\n<p style=\"position:absolute;top:290px;left:160px;white-space:nowrap\" class=\"ft1210\">&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:661px;left:108px;white-space:nowrap\" class=\"ft123\">Let&nbsp;a&nbsp;photon&nbsp;of&nbsp;energy &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;collides&nbsp;with an&nbsp;electron&nbsp;at&nbsp;rest.&nbsp;During&nbsp;the&nbsp;collision it&nbsp;gives&nbsp;a&nbsp;small fraction&nbsp;<\/p>\n<p style=\"position:absolute;top:681px;left:108px;white-space:nowrap\" class=\"ft123\">of&nbsp;energy&nbsp;to&nbsp;the&nbsp;frequency&nbsp;of&nbsp;electron.&nbsp;The&nbsp;electron&nbsp;gains&nbsp;kinetic&nbsp;energy&nbsp;and&nbsp;recoils.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:701px;left:108px;white-space:nowrap\" class=\"ft123\">&nbsp;<\/p>\n<p style=\"position:absolute;top:724px;left:135px;white-space:nowrap\" class=\"ft123\">&#61591;&nbsp;<\/p>\n<p style=\"position:absolute;top:721px;left:162px;white-space:nowrap\" class=\"ft124\"><b>Before collision&nbsp;&nbsp;<\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:760px;left:127px;white-space:nowrap\" class=\"ft123\">i.&nbsp;<\/p>\n<p style=\"position:absolute;top:760px;left:162px;white-space:nowrap\" class=\"ft123\">Energy&nbsp;of&nbsp;incident&nbsp;photon&nbsp;&nbsp;=&nbsp;<\/p>\n<p style=\"position:absolute;top:798px;left:123px;white-space:nowrap\" class=\"ft123\">ii.&nbsp;<\/p>\n<p style=\"position:absolute;top:798px;left:162px;white-space:nowrap\" class=\"ft123\">Momentum&nbsp;of&nbsp; incident&nbsp;photon=&nbsp;<\/p>\n<p style=\"position:absolute;top:836px;left:119px;white-space:nowrap\" class=\"ft123\">iii.&nbsp;<\/p>\n<p style=\"position:absolute;top:836px;left:162px;white-space:nowrap\" class=\"ft123\">Rest&nbsp;mass&nbsp;of&nbsp;free&nbsp;electron=&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:874px;left:120px;white-space:nowrap\" class=\"ft123\">iv.&nbsp;<\/p>\n<p style=\"position:absolute;top:874px;left:162px;white-space:nowrap\" class=\"ft123\">Momentum&nbsp;of&nbsp;rest electron=0&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:915px;left:135px;white-space:nowrap\" class=\"ft123\">&#61591;&nbsp;<\/p>\n<p style=\"position:absolute;top:912px;left:162px;white-space:nowrap\" class=\"ft123\">After&nbsp;collision&nbsp;<\/p>\n<p style=\"position:absolute;top:950px;left:127px;white-space:nowrap\" class=\"ft123\">i.&nbsp;<\/p>\n<p style=\"position:absolute;top:950px;left:162px;white-space:nowrap\" class=\"ft123\">Energy&nbsp;of&nbsp;scattered&nbsp;photon&nbsp;=&nbsp;<\/p>\n<p style=\"position:absolute;top:989px;left:123px;white-space:nowrap\" class=\"ft123\">ii.&nbsp;<\/p>\n<p style=\"position:absolute;top:989px;left:162px;white-space:nowrap\" class=\"ft123\">Momentum of&nbsp;scattered&nbsp;photon=&nbsp;<\/p>\n<p style=\"position:absolute;top:1027px;left:119px;white-space:nowrap\" class=\"ft123\">iii.&nbsp;<\/p>\n<p style=\"position:absolute;top:1027px;left:162px;white-space:nowrap\" class=\"ft123\">Energy&nbsp;of&nbsp;electron=&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:1065px;left:120px;white-space:nowrap\" class=\"ft123\">iv.&nbsp;<\/p>\n<p style=\"position:absolute;top:1065px;left:162px;white-space:nowrap\" class=\"ft123\">Momentum&nbsp;of&nbsp;recoil electron=mv&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:1103px;left:135px;white-space:nowrap\" class=\"ft123\">&nbsp;<\/p>\n<p style=\"position:absolute;top:756px;left:387px;white-space:nowrap\" class=\"ft120\">&nu;<\/p>\n<p style=\"position:absolute;top:763px;left:378px;white-space:nowrap\" class=\"ft125\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:818px;left:404px;white-space:nowrap\" class=\"ft126\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:792px;left:397px;white-space:nowrap\" class=\"ft126\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:785px;left:406px;white-space:nowrap\" class=\"ft120\">&nu;<\/p>\n<p style=\"position:absolute;top:838px;left:384px;white-space:nowrap\" class=\"ft127\">2<\/p>\n<p style=\"position:absolute;top:851px;left:369px;white-space:nowrap\" class=\"ft127\">0<\/p>\n<p style=\"position:absolute;top:841px;left:375px;white-space:nowrap\" class=\"ft125\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:841px;left:357px;white-space:nowrap\" class=\"ft125\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:958px;left:396px;white-space:nowrap\" class=\"ft128\">&#8216;<\/p>\n<p style=\"position:absolute;top:951px;left:384px;white-space:nowrap\" class=\"ft120\">&nu;<\/p>\n<p style=\"position:absolute;top:958px;left:375px;white-space:nowrap\" class=\"ft125\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:1005px;left:421px;white-space:nowrap\" class=\"ft126\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:980px;left:414px;white-space:nowrap\" class=\"ft126\"><i>h&nbsp;<\/i>&#8216;<\/p>\n<p style=\"position:absolute;top:973px;left:423px;white-space:nowrap\" class=\"ft120\">&nu;<\/p>\n<p style=\"position:absolute;top:1030px;left:329px;white-space:nowrap\" class=\"ft127\">2<\/p>\n<p style=\"position:absolute;top:1033px;left:307px;white-space:nowrap\" class=\"ft125\"><i>mc<\/i><\/p>\n<p style=\"position:absolute;top:658px;left:280px;white-space:nowrap\" class=\"ft120\">&nu;<\/p>\n<p style=\"position:absolute;top:665px;left:271px;white-space:nowrap\" class=\"ft125\"><i>h<\/i><\/p>\n<\/div>\n<div id=\"page13-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224013.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft130\">12&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft130\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:66px;left:108px;white-space:nowrap\" class=\"ft131\">Where&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:87px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:107px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:127px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:147px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:170px;left:135px;white-space:nowrap\" class=\"ft131\">&#61591;&nbsp;<\/p>\n<p style=\"position:absolute;top:167px;left:162px;white-space:nowrap\" class=\"ft131\">Energy&nbsp;of&nbsp;system&nbsp;before&nbsp;collision=&nbsp;<\/p>\n<p style=\"position:absolute;top:208px;left:135px;white-space:nowrap\" class=\"ft131\">&#61591;&nbsp;<\/p>\n<p style=\"position:absolute;top:205px;left:162px;white-space:nowrap\" class=\"ft131\">Energy&nbsp;of&nbsp;system&nbsp;after&nbsp;collision=&nbsp;<\/p>\n<p style=\"position:absolute;top:246px;left:135px;white-space:nowrap\" class=\"ft131\">&#61591;&nbsp;<\/p>\n<p style=\"position:absolute;top:243px;left:162px;white-space:nowrap\" class=\"ft131\">Momentum&nbsp;before&nbsp;collision=&nbsp;momentum&nbsp;after&nbsp;collision&nbsp;<\/p>\n<p style=\"position:absolute;top:281px;left:162px;white-space:nowrap\" class=\"ft130\">&nbsp;<\/p>\n<p style=\"position:absolute;top:303px;left:162px;white-space:nowrap\" class=\"ft130\">&nbsp;<\/p>\n<p style=\"position:absolute;top:325px;left:162px;white-space:nowrap\" class=\"ft130\">Where,&nbsp;<i>h<\/i>&nbsp;i<a href=\"http:\/\/en.wikipedia.org\/wiki\/Planck%27s_constant\" target=\"_blank\" rel=\"noopener\">s&nbsp;Planck&#8217;s&nbsp;constant.<\/a>&nbsp;<\/p>\n<p style=\"position:absolute;top:368px;left:108px;white-space:nowrap\" class=\"ft130\">Before&nbsp;the&nbsp;scattering&nbsp;event, the&nbsp;electron is&nbsp;treated as&nbsp;sufficiently&nbsp;close&nbsp;to&nbsp;being&nbsp;at rest that its&nbsp;<\/p>\n<p style=\"position:absolute;top:390px;left:108px;white-space:nowrap\" class=\"ft130\">total&nbsp;energy&nbsp;consists&nbsp;entirely&nbsp;of the&nbsp;mass-energy&nbsp;equivalence&nbsp;of its&nbsp;rest&nbsp;mass&nbsp;<\/p>\n<p style=\"position:absolute;top:392px;left:671px;white-space:nowrap\" class=\"ft131\">m<\/p>\n<p style=\"position:absolute;top:390px;left:684px;white-space:nowrap\" class=\"ft130\">:&nbsp;<\/p>\n<p style=\"position:absolute;top:440px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:439px;left:361px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:442px;left:339px;white-space:nowrap\" class=\"ft134\"><i>mc<\/i><\/p>\n<p style=\"position:absolute;top:442px;left:309px;white-space:nowrap\" class=\"ft134\"><i>E<\/i><\/p>\n<p style=\"position:absolute;top:436px;left:325px;white-space:nowrap\" class=\"ft135\">=<\/p>\n<p style=\"position:absolute;top:435px;left:370px;white-space:nowrap\" class=\"ft130\">&nbsp;<\/p>\n<p style=\"position:absolute;top:478px;left:108px;white-space:nowrap\" class=\"ft130\">&nbsp;<\/p>\n<p style=\"position:absolute;top:522px;left:108px;white-space:nowrap\" class=\"ft131\">Squaring&nbsp;and&nbsp;adding&nbsp;above&nbsp;eqs.&nbsp;<\/p>\n<p style=\"position:absolute;top:525px;left:380px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:542px;left:162px;white-space:nowrap\" class=\"ft130\">&nbsp;<\/p>\n<p style=\"position:absolute;top:585px;left:108px;white-space:nowrap\" class=\"ft130\">&nbsp;<\/p>\n<p style=\"position:absolute;top:628px;left:108px;white-space:nowrap\" class=\"ft130\">&nbsp;<\/p>\n<p style=\"position:absolute;top:651px;left:108px;white-space:nowrap\" class=\"ft138\">According to&nbsp;the principle&nbsp;of&nbsp;conservation&nbsp;of&nbsp;energy,&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:700px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&#119864;&#119864;&nbsp;=&nbsp;&#8462;(<\/p>\n<p style=\"position:absolute;top:696px;left:291px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:700px;left:309px;white-space:nowrap\" class=\"ft131\">&minus;&nbsp;&prime;)&nbsp;+<\/p>\n<p style=\"position:absolute;top:701px;left:404px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:713px;left:389px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:704px;left:395px;white-space:nowrap\" class=\"ft134\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:704px;left:377px;white-space:nowrap\" class=\"ft134\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:702px;left:411px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:737px;left:108px;white-space:nowrap\" class=\"ft131\">From&nbsp;relativistic&nbsp;mechanics,&nbsp;&#119864;&#119864;&nbsp;=&nbsp;&#65533;<\/p>\n<p style=\"position:absolute;top:739px;left:373px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:739px;left:358px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:742px;left:364px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:742px;left:347px;white-space:nowrap\" class=\"ft132\"><i>p<\/i><\/p>\n<p style=\"position:absolute;top:738px;left:385px;white-space:nowrap\" class=\"ft131\">+&nbsp;m<\/p>\n<p style=\"position:absolute;top:747px;left:415px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:736px;left:415px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:738px;left:423px;white-space:nowrap\" class=\"ft131\">c<\/p>\n<p style=\"position:absolute;top:737px;left:431px;white-space:nowrap\" class=\"ft137\">4<\/p>\n<p style=\"position:absolute;top:737px;left:438px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:774px;left:108px;white-space:nowrap\" class=\"ft131\">On&nbsp;comparing&nbsp;above eq.&nbsp;,we&nbsp;get&nbsp;&nbsp;&#65533;&#8462;(<\/p>\n<p style=\"position:absolute;top:771px;left:361px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:775px;left:378px;white-space:nowrap\" class=\"ft131\">&minus;<\/p>\n<p style=\"position:absolute;top:771px;left:412px;white-space:nowrap\" class=\"ft137\">&prime;<\/p>\n<p style=\"position:absolute;top:775px;left:417px;white-space:nowrap\" class=\"ft131\">)&nbsp;+<\/p>\n<p style=\"position:absolute;top:776px;left:473px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:788px;left:459px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:779px;left:464px;white-space:nowrap\" class=\"ft134\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:779px;left:446px;white-space:nowrap\" class=\"ft134\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:775px;left:481px;white-space:nowrap\" class=\"ft131\">&#65533;<\/p>\n<p style=\"position:absolute;top:764px;left:489px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:775px;left:501px;white-space:nowrap\" class=\"ft131\">=<\/p>\n<p style=\"position:absolute;top:775px;left:548px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:775px;left:533px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:778px;left:539px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:778px;left:523px;white-space:nowrap\" class=\"ft132\"><i>p<\/i><\/p>\n<p style=\"position:absolute;top:775px;left:561px;white-space:nowrap\" class=\"ft131\">+&nbsp;m<\/p>\n<p style=\"position:absolute;top:783px;left:591px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:773px;left:591px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:775px;left:598px;white-space:nowrap\" class=\"ft131\">c<\/p>\n<p style=\"position:absolute;top:773px;left:606px;white-space:nowrap\" class=\"ft137\">4<\/p>\n<p style=\"position:absolute;top:776px;left:614px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:811px;left:266px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:811px;left:214px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:811px;left:180px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:817px;left:257px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:817px;left:252px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:817px;left:224px;white-space:nowrap\" class=\"ft130\">2<\/p>\n<p style=\"position:absolute;top:817px;left:203px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:817px;left:154px;white-space:nowrap\" class=\"ft130\">(<\/p>\n<p style=\"position:absolute;top:814px;left:207px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:814px;left:170px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:814px;left:147px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:810px;left:230px;white-space:nowrap\" class=\"ft130\">&nu;&nu;<\/p>\n<p style=\"position:absolute;top:810px;left:191px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:817px;left:161px;white-space:nowrap\" class=\"ft132\"><i>v<\/i><\/p>\n<p style=\"position:absolute;top:817px;left:138px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:814px;left:279px;white-space:nowrap\" class=\"ft131\">m<\/p>\n<p style=\"position:absolute;top:822px;left:293px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:811px;left:293px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:814px;left:300px;white-space:nowrap\" class=\"ft131\">c<\/p>\n<p style=\"position:absolute;top:811px;left:307px;white-space:nowrap\" class=\"ft137\">4<\/p>\n<p style=\"position:absolute;top:814px;left:319px;white-space:nowrap\" class=\"ft131\">+&nbsp;2h(<\/p>\n<p style=\"position:absolute;top:810px;left:360px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:814px;left:378px;white-space:nowrap\" class=\"ft131\">&minus;<\/p>\n<p style=\"position:absolute;top:810px;left:394px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:814px;left:407px;white-space:nowrap\" class=\"ft131\">&prime;)<\/p>\n<p style=\"position:absolute;top:814px;left:448px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:827px;left:434px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:817px;left:439px;white-space:nowrap\" class=\"ft134\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:817px;left:421px;white-space:nowrap\" class=\"ft134\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:814px;left:461px;white-space:nowrap\" class=\"ft131\">=<\/p>\n<p style=\"position:absolute;top:814px;left:509px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:814px;left:493px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:817px;left:499px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:817px;left:483px;white-space:nowrap\" class=\"ft132\"><i>p<\/i><\/p>\n<p style=\"position:absolute;top:814px;left:521px;white-space:nowrap\" class=\"ft131\">+&nbsp;m<\/p>\n<p style=\"position:absolute;top:822px;left:551px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:811px;left:551px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:814px;left:559px;white-space:nowrap\" class=\"ft131\">c<\/p>\n<p style=\"position:absolute;top:811px;left:566px;white-space:nowrap\" class=\"ft137\">4<\/p>\n<p style=\"position:absolute;top:808px;left:573px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:833px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:858px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:859px;left:160px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:859px;left:145px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:862px;left:151px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:862px;left:135px;white-space:nowrap\" class=\"ft132\"><i>p<\/i><\/p>\n<p style=\"position:absolute;top:858px;left:174px;white-space:nowrap\" class=\"ft131\">=<\/p>\n<p style=\"position:absolute;top:856px;left:322px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:856px;left:270px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:856px;left:236px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:862px;left:313px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:862px;left:308px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:862px;left:280px;white-space:nowrap\" class=\"ft130\">2<\/p>\n<p style=\"position:absolute;top:862px;left:259px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:862px;left:210px;white-space:nowrap\" class=\"ft130\">(<\/p>\n<p style=\"position:absolute;top:859px;left:263px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:859px;left:226px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:859px;left:203px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:855px;left:286px;white-space:nowrap\" class=\"ft130\">&nu;&nu;<\/p>\n<p style=\"position:absolute;top:855px;left:247px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:862px;left:217px;white-space:nowrap\" class=\"ft132\"><i>v<\/i><\/p>\n<p style=\"position:absolute;top:862px;left:193px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:858px;left:335px;white-space:nowrap\" class=\"ft131\">2h(<\/p>\n<p style=\"position:absolute;top:855px;left:360px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:858px;left:377px;white-space:nowrap\" class=\"ft131\">&minus;<\/p>\n<p style=\"position:absolute;top:855px;left:393px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:858px;left:406px;white-space:nowrap\" class=\"ft131\">&prime;)<\/p>\n<p style=\"position:absolute;top:859px;left:447px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:871px;left:433px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:862px;left:438px;white-space:nowrap\" class=\"ft134\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:862px;left:420px;white-space:nowrap\" class=\"ft134\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:860px;left:455px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:900px;left:520px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:900px;left:516px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:900px;left:482px;white-space:nowrap\" class=\"ft130\">)(<\/p>\n<p style=\"position:absolute;top:900px;left:454px;white-space:nowrap\" class=\"ft130\">(<\/p>\n<p style=\"position:absolute;top:900px;left:445px;white-space:nowrap\" class=\"ft130\">2<\/p>\n<p style=\"position:absolute;top:900px;left:425px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:900px;left:324px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:900px;left:288px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:900px;left:280px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:900px;left:276px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:900px;left:242px;white-space:nowrap\" class=\"ft130\">)(<\/p>\n<p style=\"position:absolute;top:900px;left:214px;white-space:nowrap\" class=\"ft130\">(<\/p>\n<p style=\"position:absolute;top:900px;left:205px;white-space:nowrap\" class=\"ft130\">2<\/p>\n<p style=\"position:absolute;top:900px;left:185px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:897px;left:428px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:897px;left:409px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:897px;left:375px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:897px;left:360px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:897px;left:188px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:897px;left:169px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:897px;left:135px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:897px;left:121px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:894px;left:503px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:894px;left:470px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:894px;left:412px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:894px;left:313px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:894px;left:264px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:894px;left:230px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:894px;left:172px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:900px;left:494px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:461px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:399px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:366px;white-space:nowrap\" class=\"ft132\"><i>v<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:351px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:255px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:221px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:159px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:126px;white-space:nowrap\" class=\"ft132\"><i>v<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:111px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:895px;left:435px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:895px;left:385px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:895px;left:336px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:895px;left:196px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:895px;left:146px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:897px;left:534px;white-space:nowrap\" class=\"ft131\">+&nbsp;2h<\/p>\n<p style=\"position:absolute;top:893px;left:568px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:898px;left:612px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:910px;left:597px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:901px;left:603px;white-space:nowrap\" class=\"ft134\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:901px;left:585px;white-space:nowrap\" class=\"ft134\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:897px;left:623px;white-space:nowrap\" class=\"ft131\">&minus;&nbsp;2h<\/p>\n<p style=\"position:absolute;top:893px;left:658px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:897px;left:671px;white-space:nowrap\" class=\"ft131\">&prime;<\/p>\n<p style=\"position:absolute;top:898px;left:705px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:910px;left:691px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:901px;left:696px;white-space:nowrap\" class=\"ft134\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:901px;left:678px;white-space:nowrap\" class=\"ft134\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:900px;left:713px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:917px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:935px;left:285px;white-space:nowrap\" class=\"ft131\">(&#8462;&#120584;&#120584;)(&#8462;&#120584;&#120584;<\/p>\n<p style=\"position:absolute;top:934px;left:342px;white-space:nowrap\" class=\"ft137\">&prime;<\/p>\n<p style=\"position:absolute;top:935px;left:347px;white-space:nowrap\" class=\"ft131\">)&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&nbsp;=&nbsp;(&#8462;&#120584;&#120584;)(&#8462;&#120584;&#120584;<\/p>\n<p style=\"position:absolute;top:934px;left:467px;white-space:nowrap\" class=\"ft137\">&prime;<\/p>\n<p style=\"position:absolute;top:935px;left:472px;white-space:nowrap\" class=\"ft131\">)&nbsp;&minus;&nbsp;&#8462;&#120584;&#120584;&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:944px;left:530px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:936px;left:538px;white-space:nowrap\" class=\"ft131\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:934px;left:546px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:936px;left:557px;white-space:nowrap\" class=\"ft131\">+&nbsp;&#8462;&#120584;&#120584;&prime;&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:944px;left:609px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:936px;left:617px;white-space:nowrap\" class=\"ft131\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:934px;left:625px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:937px;left:633px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:954px;left:108px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:975px;left:108px;white-space:nowrap\" class=\"ft131\">Dividing&nbsp;the&nbsp;above eq.&nbsp;by&nbsp;&nbsp;(&#8462;&#120584;&#120584;)(&#8462;&#120584;&#120584;<\/p>\n<p style=\"position:absolute;top:975px;left:351px;white-space:nowrap\" class=\"ft137\">&prime;<\/p>\n<p style=\"position:absolute;top:976px;left:356px;white-space:nowrap\" class=\"ft131\">),&nbsp;we&nbsp;get&nbsp;<\/p>\n<p style=\"position:absolute;top:1007px;left:347px;white-space:nowrap\" class=\"ft131\">&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&nbsp;=&nbsp;1&nbsp;&minus;<\/p>\n<p style=\"position:absolute;top:995px;left:437px;white-space:nowrap\" class=\"ft131\">1<\/p>\n<p style=\"position:absolute;top:1018px;left:431px;white-space:nowrap\" class=\"ft131\">&#8462;&#120584;&#120584;&prime;&nbsp;&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:1015px;left:469px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:1007px;left:477px;white-space:nowrap\" class=\"ft131\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:1005px;left:485px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:1007px;left:497px;white-space:nowrap\" class=\"ft131\">+<\/p>\n<p style=\"position:absolute;top:995px;left:517px;white-space:nowrap\" class=\"ft131\">1<\/p>\n<p style=\"position:absolute;top:1018px;left:513px;white-space:nowrap\" class=\"ft131\">&#8462;&#120584;&#120584;&nbsp;&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:1015px;left:547px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:1007px;left:555px;white-space:nowrap\" class=\"ft131\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:1005px;left:563px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:1009px;left:571px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1032px;left:353px;white-space:nowrap\" class=\"ft131\">1<\/p>\n<p style=\"position:absolute;top:1055px;left:347px;white-space:nowrap\" class=\"ft131\">&#8462;&#120584;&#120584;&prime;&nbsp;&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:1052px;left:385px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:1044px;left:393px;white-space:nowrap\" class=\"ft131\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:1043px;left:401px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:1044px;left:413px;white-space:nowrap\" class=\"ft131\">&minus;<\/p>\n<p style=\"position:absolute;top:1032px;left:433px;white-space:nowrap\" class=\"ft131\">1<\/p>\n<p style=\"position:absolute;top:1055px;left:429px;white-space:nowrap\" class=\"ft131\">&#8462;&#120584;&#120584;&nbsp;&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:1052px;left:463px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:1044px;left:471px;white-space:nowrap\" class=\"ft131\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:1043px;left:479px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:1044px;left:492px;white-space:nowrap\" class=\"ft131\">=&nbsp;1&nbsp;&minus;&nbsp;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&nbsp;<\/p>\n<p style=\"position:absolute;top:1069px;left:371px;white-space:nowrap\" class=\"ft131\">1<\/p>\n<p style=\"position:absolute;top:1093px;left:369px;white-space:nowrap\" class=\"ft131\">&#120584;&#120584;&prime;&nbsp;&minus;<\/p>\n<p style=\"position:absolute;top:1069px;left:402px;white-space:nowrap\" class=\"ft139\">1<br \/>&#120584;&#120584;&nbsp;=<\/p>\n<p style=\"position:absolute;top:1069px;left:446px;white-space:nowrap\" class=\"ft131\">&#8462;<\/p>\n<p style=\"position:absolute;top:1093px;left:432px;white-space:nowrap\" class=\"ft131\">&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:1101px;left:446px;white-space:nowrap\" class=\"ft137\">0<\/p>\n<p style=\"position:absolute;top:1093px;left:454px;white-space:nowrap\" class=\"ft131\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:1092px;left:462px;white-space:nowrap\" class=\"ft137\">2<\/p>\n<p style=\"position:absolute;top:1082px;left:473px;white-space:nowrap\" class=\"ft131\">(1&nbsp;&minus;&nbsp;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;)&nbsp;<\/p>\n<p style=\"position:absolute;top:1115px;left:459px;white-space:nowrap\" class=\"ft131\">&nbsp;<\/p>\n<p style=\"position:absolute;top:696px;left:328px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:771px;left:396px;white-space:nowrap\" class=\"ft136\">&nu;<\/p>\n<p style=\"position:absolute;top:188px;left:477px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:201px;left:462px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:191px;left:468px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:191px;left:450px;white-space:nowrap\" class=\"ft132\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:191px;left:412px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:185px;left:436px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:184px;left:421px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:209px;left:453px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:212px;left:417px;white-space:nowrap\" class=\"ft130\">&#8216;&nbsp;<i>mc<\/i><\/p>\n<p style=\"position:absolute;top:212px;left:396px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:206px;left:420px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:205px;left:405px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:244px;left:693px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:244px;left:611px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:256px;left:596px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:247px;left:658px;white-space:nowrap\" class=\"ft130\">&#8216;&nbsp;<i>mc<\/i><\/p>\n<p style=\"position:absolute;top:247px;left:636px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:247px;left:602px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:247px;left:583px;white-space:nowrap\" class=\"ft132\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:247px;left:546px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:241px;left:661px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:241px;left:622px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:241px;left:570px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:240px;left:645px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:240px;left:555px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:111px;left:268px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:86px;left:268px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:70px;left:254px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:100px;left:234px;white-space:nowrap\" class=\"ft130\">1<\/p>\n<p style=\"position:absolute;top:114px;left:259px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:89px;left:259px;white-space:nowrap\" class=\"ft132\"><i>v<\/i><\/p>\n<p style=\"position:absolute;top:60px;left:241px;white-space:nowrap\" class=\"ft132\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:71px;left:191px;white-space:nowrap\" class=\"ft132\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:94px;left:244px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:66px;left:208px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:277px;left:489px;white-space:nowrap\" class=\"ft130\">&phi;<\/p>\n<p style=\"position:absolute;top:277px;left:413px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:266px;left:356px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:266px;left:289px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:284px;left:464px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:284px;left:388px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:273px;left:368px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:284px;left:319px;white-space:nowrap\" class=\"ft130\">0<\/p>\n<p style=\"position:absolute;top:284px;left:441px;white-space:nowrap\" class=\"ft132\"><i>mv<\/i><\/p>\n<p style=\"position:absolute;top:298px;left:355px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:273px;left:347px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:298px;left:286px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:273px;left:280px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:278px;left:428px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:278px;left:376px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:278px;left:331px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:278px;left:306px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:459px;left:428px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:459px;left:387px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:459px;left:313px;white-space:nowrap\" class=\"ft130\">&phi;<\/p>\n<p style=\"position:absolute;top:465px;left:404px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:465px;left:399px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:465px;left:288px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:465px;left:378px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:465px;left:344px;white-space:nowrap\" class=\"ft132\"><i>hv<\/i><\/p>\n<p style=\"position:absolute;top:465px;left:269px;white-space:nowrap\" class=\"ft132\"><i>pc<\/i><\/p>\n<p style=\"position:absolute;top:460px;left:365px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:460px;left:329px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:486px;left:389px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:486px;left:350px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:486px;left:311px;white-space:nowrap\" class=\"ft130\">&phi;<\/p>\n<p style=\"position:absolute;top:493px;left:367px;white-space:nowrap\" class=\"ft130\">sin<\/p>\n<p style=\"position:absolute;top:493px;left:363px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:493px;left:288px;white-space:nowrap\" class=\"ft130\">sin<\/p>\n<p style=\"position:absolute;top:493px;left:341px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:493px;left:269px;white-space:nowrap\" class=\"ft132\"><i>pc<\/i><\/p>\n<p style=\"position:absolute;top:487px;left:326px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:550px;left:490px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:550px;left:394px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:550px;left:258px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:550px;left:243px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:553px;left:483px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:553px;left:449px;white-space:nowrap\" class=\"ft130\">sin<\/p>\n<p style=\"position:absolute;top:553px;left:445px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:553px;left:417px;white-space:nowrap\" class=\"ft130\">(<\/p>\n<p style=\"position:absolute;top:553px;left:387px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:553px;left:351px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:553px;left:346px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:553px;left:284px;white-space:nowrap\" class=\"ft130\">(<\/p>\n<p style=\"position:absolute;top:546px;left:471px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:546px;left:433px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:546px;left:375px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:546px;left:334px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:553px;left:424px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:553px;left:325px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:553px;left:291px;white-space:nowrap\" class=\"ft132\"><i>hv<\/i><\/p>\n<p style=\"position:absolute;top:553px;left:249px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:553px;left:233px;white-space:nowrap\" class=\"ft132\"><i>p<\/i><\/p>\n<p style=\"position:absolute;top:547px;left:404px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:547px;left:311px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:547px;left:270px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:615px;left:500px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:615px;left:464px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:615px;left:456px;white-space:nowrap\" class=\"ft130\">)<\/p>\n<p style=\"position:absolute;top:615px;left:452px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:615px;left:418px;white-space:nowrap\" class=\"ft130\">)(<\/p>\n<p style=\"position:absolute;top:615px;left:390px;white-space:nowrap\" class=\"ft130\">(<\/p>\n<p style=\"position:absolute;top:615px;left:381px;white-space:nowrap\" class=\"ft130\">2<\/p>\n<p style=\"position:absolute;top:615px;left:361px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:586px;left:572px;white-space:nowrap\" class=\"ft130\">sin<\/p>\n<p style=\"position:absolute;top:586px;left:560px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:586px;left:474px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:586px;left:462px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:586px;left:384px;white-space:nowrap\" class=\"ft130\">cos<\/p>\n<p style=\"position:absolute;top:586px;left:379px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:586px;left:335px;white-space:nowrap\" class=\"ft130\">2<\/p>\n<p style=\"position:absolute;top:612px;left:364px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:612px;left:345px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:612px;left:311px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:612px;left:296px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:595px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:564px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:545px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:499px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:466px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:447px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:354px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:311px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:583px;left:296px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:608px;left:488px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:608px;left:439px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:608px;left:406px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:608px;left:348px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:579px;left:602px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:579px;left:548px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:579px;left:506px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:579px;left:450px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:579px;left:408px;white-space:nowrap\" class=\"ft130\">&theta;<\/p>\n<p style=\"position:absolute;top:579px;left:357px;white-space:nowrap\" class=\"ft130\">&nu;&nu;<\/p>\n<p style=\"position:absolute;top:615px;left:430px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:615px;left:397px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:615px;left:335px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:615px;left:302px;white-space:nowrap\" class=\"ft132\"><i>v<\/i><\/p>\n<p style=\"position:absolute;top:615px;left:287px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:586px;left:535px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:586px;left:437px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:586px;left:344px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:586px;left:302px;white-space:nowrap\" class=\"ft132\"><i>v<\/i><\/p>\n<p style=\"position:absolute;top:586px;left:287px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:609px;left:371px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:609px;left:322px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:609px;left:272px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:580px;left:521px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:580px;left:423px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:580px;left:322px;white-space:nowrap\" class=\"ft130\">&minus;<\/p>\n<p style=\"position:absolute;top:580px;left:272px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:677px;left:368px;white-space:nowrap\" class=\"ft132\"><i>E<\/i><\/p>\n<p style=\"position:absolute;top:677px;left:332px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:677px;left:296px;white-space:nowrap\" class=\"ft132\"><i>c<\/i><\/p>\n<p style=\"position:absolute;top:677px;left:278px;white-space:nowrap\" class=\"ft132\"><i>m<\/i><\/p>\n<p style=\"position:absolute;top:677px;left:240px;white-space:nowrap\" class=\"ft132\"><i>h<\/i><\/p>\n<p style=\"position:absolute;top:672px;left:357px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:672px;left:317px;white-space:nowrap\" class=\"ft130\">=<\/p>\n<p style=\"position:absolute;top:672px;left:264px;white-space:nowrap\" class=\"ft130\">+<\/p>\n<p style=\"position:absolute;top:677px;left:354px;white-space:nowrap\" class=\"ft130\">&#8216;<\/p>\n<p style=\"position:absolute;top:675px;left:305px;white-space:nowrap\" class=\"ft133\">2<\/p>\n<p style=\"position:absolute;top:687px;left:291px;white-space:nowrap\" class=\"ft133\">0<\/p>\n<p style=\"position:absolute;top:671px;left:341px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<p style=\"position:absolute;top:671px;left:249px;white-space:nowrap\" class=\"ft130\">&nu;<\/p>\n<\/div>\n<div id=\"page14-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224014.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft140\">13&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:50px;left:459px;white-space:nowrap\" class=\"ft141\">&nbsp;<\/p>\n<p style=\"position:absolute;top:76px;left:335px;white-space:nowrap\" class=\"ft141\">&#119888;&#119888;&nbsp;=&nbsp;&#120584;&#120584;<\/p>\n<p style=\"position:absolute;top:74px;left:373px;white-space:nowrap\" class=\"ft142\">&lambda;<\/p>\n<p style=\"position:absolute;top:76px;left:382px;white-space:nowrap\" class=\"ft141\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&#119888;&#119888;&#119900;&#119900;&nbsp;&nbsp;&#120584;&#120584;&nbsp;=<\/p>\n<p style=\"position:absolute;top:63px;left:477px;white-space:nowrap\" class=\"ft141\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:85px;left:473px;white-space:nowrap\" class=\"ft142\">&lambda;<\/p>\n<p style=\"position:absolute;top:87px;left:482px;white-space:nowrap\" class=\"ft141\">&nbsp;&nbsp;&nbsp;&nbsp;&#119886;&#119886;&#119886;&#119886;&#119886;&#119886;&nbsp;&#120584;&#120584;&prime;&nbsp;=<\/p>\n<p style=\"position:absolute;top:63px;left:569px;white-space:nowrap\" class=\"ft141\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:85px;left:562px;white-space:nowrap\" class=\"ft142\">&lambda;<\/p>\n<p style=\"position:absolute;top:87px;left:571px;white-space:nowrap\" class=\"ft141\">&nbsp;&prime;&nbsp;<\/p>\n<p style=\"position:absolute;top:77px;left:583px;white-space:nowrap\" class=\"ft141\">&nbsp;<\/p>\n<p style=\"position:absolute;top:100px;left:369px;white-space:nowrap\" class=\"ft142\">&lambda;<\/p>\n<p style=\"position:absolute;top:102px;left:378px;white-space:nowrap\" class=\"ft141\">&prime;<\/p>\n<p style=\"position:absolute;top:126px;left:372px;white-space:nowrap\" class=\"ft141\">&#119888;&#119888;&nbsp;&minus;<\/p>\n<p style=\"position:absolute;top:100px;left:402px;white-space:nowrap\" class=\"ft142\">&lambda;<\/p>\n<p style=\"position:absolute;top:126px;left:402px;white-space:nowrap\" class=\"ft141\">&#119888;&#119888;&nbsp;=<\/p>\n<p style=\"position:absolute;top:102px;left:446px;white-space:nowrap\" class=\"ft141\">&#8462;<\/p>\n<p style=\"position:absolute;top:126px;left:433px;white-space:nowrap\" class=\"ft141\">&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:133px;left:446px;white-space:nowrap\" class=\"ft143\">0<\/p>\n<p style=\"position:absolute;top:126px;left:454px;white-space:nowrap\" class=\"ft141\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:125px;left:462px;white-space:nowrap\" class=\"ft143\">2<\/p>\n<p style=\"position:absolute;top:115px;left:473px;white-space:nowrap\" class=\"ft141\">(1&nbsp;&minus;&nbsp;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;)&nbsp;<\/p>\n<p style=\"position:absolute;top:154px;left:373px;white-space:nowrap\" class=\"ft142\">&lambda;<\/p>\n<p style=\"position:absolute;top:152px;left:382px;white-space:nowrap\" class=\"ft143\">&prime;<\/p>\n<p style=\"position:absolute;top:156px;left:390px;white-space:nowrap\" class=\"ft141\">&minus;<\/p>\n<p style=\"position:absolute;top:154px;left:406px;white-space:nowrap\" class=\"ft142\">&lambda;<\/p>\n<p style=\"position:absolute;top:156px;left:420px;white-space:nowrap\" class=\"ft141\">=<\/p>\n<p style=\"position:absolute;top:143px;left:447px;white-space:nowrap\" class=\"ft141\">&#8462;<\/p>\n<p style=\"position:absolute;top:167px;left:437px;white-space:nowrap\" class=\"ft141\">&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:174px;left:451px;white-space:nowrap\" class=\"ft143\">0<\/p>\n<p style=\"position:absolute;top:167px;left:458px;white-space:nowrap\" class=\"ft141\">&#119888;&#119888;&nbsp;(1&nbsp;&minus;&nbsp;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;)<\/p>\n<p style=\"position:absolute;top:157px;left:545px;white-space:nowrap\" class=\"ft141\">&nbsp;<\/p>\n<p style=\"position:absolute;top:196px;left:381px;white-space:nowrap\" class=\"ft141\">&#9651;<\/p>\n<p style=\"position:absolute;top:195px;left:398px;white-space:nowrap\" class=\"ft142\">&lambda;<\/p>\n<p style=\"position:absolute;top:196px;left:412px;white-space:nowrap\" class=\"ft141\">=<\/p>\n<p style=\"position:absolute;top:184px;left:439px;white-space:nowrap\" class=\"ft141\">&#8462;<\/p>\n<p style=\"position:absolute;top:208px;left:429px;white-space:nowrap\" class=\"ft141\">&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:215px;left:442px;white-space:nowrap\" class=\"ft143\">0<\/p>\n<p style=\"position:absolute;top:208px;left:450px;white-space:nowrap\" class=\"ft141\">&#119888;&#119888;&nbsp;(1&nbsp;&minus;&nbsp;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;&#119888;)<\/p>\n<p style=\"position:absolute;top:198px;left:537px;white-space:nowrap\" class=\"ft141\">&nbsp;<\/p>\n<p style=\"position:absolute;top:225px;left:108px;white-space:nowrap\" class=\"ft140\">&#8710;&nbsp;&lambda;<\/p>\n<p style=\"position:absolute;top:225px;left:132px;white-space:nowrap\" class=\"ft144\">&nbsp;<\/p>\n<p style=\"position:absolute;top:225px;left:139px;white-space:nowrap\" class=\"ft140\">is&nbsp;known as&nbsp;<\/p>\n<p style=\"position:absolute;top:225px;left:227px;white-space:nowrap\" class=\"ft145\"><b>Compton shift.&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:247px;left:108px;white-space:nowrap\" class=\"ft145\"><b>Different&nbsp;cases&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:272px;left:135px;white-space:nowrap\" class=\"ft140\">&#61656;&nbsp;&nbsp;<b>If&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:271px;left:179px;white-space:nowrap\" class=\"ft140\">&#119888;&#119888;=0<\/p>\n<p style=\"position:absolute;top:271px;left:209px;white-space:nowrap\" class=\"ft143\">o<\/p>\n<p style=\"position:absolute;top:275px;left:215px;white-space:nowrap\" class=\"ft140\">,&nbsp;<\/p>\n<p style=\"position:absolute;top:269px;left:224px;white-space:nowrap\" class=\"ft140\">then&nbsp;&#8710;&nbsp;&lambda;=0.<b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:301px;left:135px;white-space:nowrap\" class=\"ft140\">&#61656;&nbsp;&nbsp;<b>If&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:300px;left:179px;white-space:nowrap\" class=\"ft140\">&#119888;&#119888;=90<\/p>\n<p style=\"position:absolute;top:301px;left:218px;white-space:nowrap\" class=\"ft143\">o<\/p>\n<p style=\"position:absolute;top:299px;left:224px;white-space:nowrap\" class=\"ft140\">, then&nbsp;&#8710;&nbsp;&lambda;=<\/p>\n<p style=\"position:absolute;top:295px;left:312px;white-space:nowrap\" class=\"ft146\">&#8462;<\/p>\n<p style=\"position:absolute;top:314px;left:303px;white-space:nowrap\" class=\"ft146\">&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:319px;left:315px;white-space:nowrap\" class=\"ft147\">&#119890;&#119890;<\/p>\n<p style=\"position:absolute;top:314px;left:322px;white-space:nowrap\" class=\"ft146\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:305px;left:328px;white-space:nowrap\" class=\"ft140\">=0.0242 A<\/p>\n<p style=\"position:absolute;top:301px;left:405px;white-space:nowrap\" class=\"ft143\">o<\/p>\n<p style=\"position:absolute;top:305px;left:411px;white-space:nowrap\" class=\"ft140\">.&nbsp;<\/p>\n<p style=\"position:absolute;top:299px;left:420px;white-space:nowrap\" class=\"ft140\">This&nbsp;constant&nbsp;value&nbsp;is&nbsp;called&nbsp;<\/p>\n<p style=\"position:absolute;top:299px;left:625px;white-space:nowrap\" class=\"ft145\"><b>Compton&nbsp;wavelength<\/b>.<b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:350px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:375px;left:135px;white-space:nowrap\" class=\"ft140\">&#61656;&nbsp;&nbsp;<b>If&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:374px;left:179px;white-space:nowrap\" class=\"ft140\">&#119888;&#119888;=180<\/p>\n<p style=\"position:absolute;top:375px;left:227px;white-space:nowrap\" class=\"ft143\">o<\/p>\n<p style=\"position:absolute;top:372px;left:233px;white-space:nowrap\" class=\"ft140\">, then &#8710;&nbsp;&lambda;=<\/p>\n<p style=\"position:absolute;top:364px;left:318px;white-space:nowrap\" class=\"ft148\">2&#8462;<\/p>\n<p style=\"position:absolute;top:387px;left:312px;white-space:nowrap\" class=\"ft148\">&#119898;&#119898;<\/p>\n<p style=\"position:absolute;top:393px;left:326px;white-space:nowrap\" class=\"ft143\">&#119890;&#119890;<\/p>\n<p style=\"position:absolute;top:387px;left:334px;white-space:nowrap\" class=\"ft148\">&#119888;&#119888;<\/p>\n<p style=\"position:absolute;top:379px;left:342px;white-space:nowrap\" class=\"ft140\">=0.0484 A<\/p>\n<p style=\"position:absolute;top:375px;left:418px;white-space:nowrap\" class=\"ft143\">o<\/p>\n<p style=\"position:absolute;top:379px;left:424px;white-space:nowrap\" class=\"ft140\">.&nbsp;<\/p>\n<p style=\"position:absolute;top:373px;left:433px;white-space:nowrap\" class=\"ft140\">This&nbsp;is&nbsp;the&nbsp;maximum&nbsp;wavelength.<\/p>\n<p style=\"position:absolute;top:372px;left:676px;white-space:nowrap\" class=\"ft145\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:422px;left:160px;white-space:nowrap\" class=\"ft149\">&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:502px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:528px;left:160px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:554px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:577px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:614px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:636px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:658px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:680px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:703px;left:108px;white-space:nowrap\" class=\"ft140\">S.NO&nbsp;<\/p>\n<p style=\"position:absolute;top:703px;left:339px;white-space:nowrap\" class=\"ft140\">RGPV QUESTIONS&nbsp;<\/p>\n<p style=\"position:absolute;top:703px;left:664px;white-space:nowrap\" class=\"ft140\">Year&nbsp;<\/p>\n<p style=\"position:absolute;top:703px;left:743px;white-space:nowrap\" class=\"ft140\">Marks&nbsp;<\/p>\n<p style=\"position:absolute;top:726px;left:108px;white-space:nowrap\" class=\"ft140\">Q.1&nbsp;<\/p>\n<p style=\"position:absolute;top:726px;left:168px;white-space:nowrap\" class=\"ft140\">What is&nbsp;Compton&nbsp;effect? Explaining&nbsp;the Compton&nbsp;expression,&nbsp;<\/p>\n<p style=\"position:absolute;top:748px;left:168px;white-space:nowrap\" class=\"ft140\">discuss&nbsp;the&nbsp;various&nbsp;possibilities&nbsp;of X-ray&nbsp;scattering?&nbsp;<\/p>\n<p style=\"position:absolute;top:726px;left:656px;white-space:nowrap\" class=\"ft140\">June&nbsp;<\/p>\n<p style=\"position:absolute;top:748px;left:656px;white-space:nowrap\" class=\"ft140\">2012&nbsp;<\/p>\n<p style=\"position:absolute;top:726px;left:720px;white-space:nowrap\" class=\"ft140\">14&nbsp;<\/p>\n<p style=\"position:absolute;top:770px;left:108px;white-space:nowrap\" class=\"ft140\">Q.2&nbsp;<\/p>\n<p style=\"position:absolute;top:770px;left:168px;white-space:nowrap\" class=\"ft140\">An X&nbsp;ray&nbsp;photon&nbsp;of Wavelength 0.4A<\/p>\n<p style=\"position:absolute;top:768px;left:451px;white-space:nowrap\" class=\"ft143\">o&nbsp;<\/p>\n<p style=\"position:absolute;top:770px;left:463px;white-space:nowrap\" class=\"ft140\">is&nbsp;scattered through an&nbsp;<\/p>\n<p style=\"position:absolute;top:792px;left:168px;white-space:nowrap\" class=\"ft140\">angle&nbsp;of&nbsp;45<\/p>\n<p style=\"position:absolute;top:790px;left:250px;white-space:nowrap\" class=\"ft143\">o<\/p>\n<p style=\"position:absolute;top:792px;left:256px;white-space:nowrap\" class=\"ft140\">&nbsp;by&nbsp;a&nbsp;loosely&nbsp;bound electron.&nbsp;Find the&nbsp;wavelength of&nbsp;<\/p>\n<p style=\"position:absolute;top:814px;left:168px;white-space:nowrap\" class=\"ft140\">the&nbsp;scattered photon.&nbsp;<\/p>\n<p style=\"position:absolute;top:770px;left:656px;white-space:nowrap\" class=\"ft1410\">DEC&nbsp;<br \/>2013&nbsp;<\/p>\n<p style=\"position:absolute;top:770px;left:720px;white-space:nowrap\" class=\"ft140\">7&nbsp;<\/p>\n<p style=\"position:absolute;top:837px;left:108px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<p style=\"position:absolute;top:837px;left:324px;white-space:nowrap\" class=\"ft140\">&nbsp;<\/p>\n<\/div>\n<div id=\"page15-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224015.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft150\">14&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft150\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:379px;white-space:nowrap\" class=\"ft151\"><b>UNIT&nbsp;1\/LECTURE&nbsp;6&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:72px;left:108px;white-space:nowrap\" class=\"ft152\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:94px;left:108px;white-space:nowrap\" class=\"ft152\"><b>&nbsp;Characteristics of&nbsp;wave function:&nbsp;[RGPV&nbsp;June&nbsp;2013 (7)]&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:116px;left:108px;white-space:nowrap\" class=\"ft150\">&nbsp;<\/p>\n<p style=\"position:absolute;top:131px;left:108px;white-space:nowrap\" class=\"ft154\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:162px;left:185px;white-space:nowrap\" class=\"ft150\">Waves&nbsp;in&nbsp;general are&nbsp;associated&nbsp;with&nbsp;quantities&nbsp;that&nbsp;vary&nbsp;periodically.&nbsp;For example,&nbsp;<\/p>\n<p style=\"position:absolute;top:193px;left:109px;white-space:nowrap\" class=\"ft155\">water&nbsp;waves&nbsp;involve&nbsp;the&nbsp;periodic&nbsp;variation of&nbsp;the&nbsp;height of the&nbsp;water&nbsp;surface&nbsp;at a&nbsp;point.&nbsp;<br \/>Similarly,&nbsp;sound&nbsp;waves&nbsp;are&nbsp;associated&nbsp;with&nbsp;periodic&nbsp;variations&nbsp;of&nbsp;the&nbsp;pressure.&nbsp;<\/p>\n<p style=\"position:absolute;top:239px;left:108px;white-space:nowrap\" class=\"ft150\">&nbsp;<\/p>\n<p style=\"position:absolute;top:257px;left:108px;white-space:nowrap\" class=\"ft150\">In&nbsp; the&nbsp; case&nbsp; of&nbsp;&nbsp;matter&nbsp;&nbsp;waves,&nbsp;&nbsp;the&nbsp;&nbsp;quantity&nbsp;&nbsp;that&nbsp;&nbsp;varies&nbsp;periodically&nbsp;is&nbsp;called&nbsp;<\/p>\n<p style=\"position:absolute;top:256px;left:690px;white-space:nowrap\" class=\"ft152\"><b>&lsquo;wave&nbsp;function&rsquo;<\/b>.&nbsp;<\/p>\n<p style=\"position:absolute;top:278px;left:108px;white-space:nowrap\" class=\"ft150\">The&nbsp;wave&nbsp;function,&nbsp;represented&nbsp;by&nbsp;&psi;,&nbsp;associated&nbsp;with&nbsp;matter&nbsp;waves&nbsp;&nbsp;has&nbsp;&nbsp;no&nbsp;&nbsp;direct&nbsp;&nbsp;physical&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:300px;left:108px;white-space:nowrap\" class=\"ft150\">significance.&nbsp; &nbsp;It &nbsp;is&nbsp; not&nbsp;an observable&nbsp;quantity.&nbsp;But the&nbsp;value&nbsp;of the&nbsp;wave&nbsp;function is&nbsp;related to&nbsp;<\/p>\n<p style=\"position:absolute;top:322px;left:108px;white-space:nowrap\" class=\"ft150\">the&nbsp;probability&nbsp;of finding&nbsp;the&nbsp;body&nbsp;at a&nbsp;given place&nbsp;at&nbsp;a&nbsp;given&nbsp;time.&nbsp;The&nbsp;square&nbsp;of&nbsp;the&nbsp;absolute&nbsp;<\/p>\n<p style=\"position:absolute;top:344px;left:108px;white-space:nowrap\" class=\"ft150\">magnitude&nbsp;of the&nbsp;wave&nbsp;function of&nbsp;a&nbsp;body&nbsp;evaluated at a&nbsp;particular&nbsp;time&nbsp;at a&nbsp;particular&nbsp;place&nbsp;is&nbsp;<\/p>\n<p style=\"position:absolute;top:366px;left:108px;white-space:nowrap\" class=\"ft150\">proportional&nbsp;to&nbsp;the&nbsp;probability&nbsp;of finding&nbsp;the&nbsp;body&nbsp;at that place&nbsp;at that instant.&nbsp;<\/p>\n<p style=\"position:absolute;top:388px;left:173px;white-space:nowrap\" class=\"ft150\">The&nbsp;wave&nbsp;functions&nbsp;are&nbsp;usually&nbsp;complex.&nbsp;&nbsp;The&nbsp;probability&nbsp;in such a&nbsp;case&nbsp;is&nbsp;taken as&nbsp;<\/p>\n<p style=\"position:absolute;top:421px;left:108px;white-space:nowrap\" class=\"ft150\">&psi;&lowast;&psi;,&nbsp;<\/p>\n<p style=\"position:absolute;top:420px;left:158px;white-space:nowrap\" class=\"ft150\">i.e.&nbsp;the&nbsp;product of the&nbsp;wave&nbsp;function with its&nbsp;complex&nbsp;conjugate.&nbsp;Since&nbsp;the&nbsp;probability&nbsp;of&nbsp;<\/p>\n<p style=\"position:absolute;top:452px;left:108px;white-space:nowrap\" class=\"ft155\">finding&nbsp;the&nbsp;body&nbsp;somewhere&nbsp;is&nbsp;finite,&nbsp;we&nbsp;have&nbsp;the&nbsp;total probability&nbsp;over all space&nbsp;equal to&nbsp;<br \/>certainty.&nbsp;<\/p>\n<p style=\"position:absolute;top:524px;left:141px;white-space:nowrap\" class=\"ft150\">i.e.<\/p>\n<p style=\"position:absolute;top:517px;left:184px;white-space:nowrap\" class=\"ft150\">&int;&nbsp;&psi;&lowast;&psi;&nbsp;dV = 1&nbsp;<\/p>\n<p style=\"position:absolute;top:544px;left:141px;white-space:nowrap\" class=\"ft150\">&nbsp;<\/p>\n<p style=\"position:absolute;top:544px;left:665px;white-space:nowrap\" class=\"ft150\">(1)<\/p>\n<p style=\"position:absolute;top:543px;left:698px;white-space:nowrap\" class=\"ft150\">&nbsp;<\/p>\n<p style=\"position:absolute;top:559px;left:108px;white-space:nowrap\" class=\"ft155\">Equation (1)&nbsp;is&nbsp;called the&nbsp;normalization condition and a&nbsp;wave&nbsp;function that obeys&nbsp;the&nbsp;equation&nbsp;<br \/>is&nbsp;said&nbsp;to&nbsp;be&nbsp;<\/p>\n<p style=\"position:absolute;top:591px;left:206px;white-space:nowrap\" class=\"ft152\"><b>normalized<\/b>.&nbsp;Further,&nbsp;&#61548;&nbsp;must be&nbsp;single&nbsp;valued since&nbsp;the&nbsp;probability&nbsp;can&nbsp;have&nbsp;only&nbsp;<\/p>\n<p style=\"position:absolute;top:623px;left:108px;white-space:nowrap\" class=\"ft156\">one&nbsp;value&nbsp;at&nbsp;a particular place&nbsp;and&nbsp;time.&nbsp;Since&nbsp;the&nbsp;probability&nbsp;can&nbsp;have&nbsp;any&nbsp;value&nbsp;between&nbsp;zero&nbsp;<br \/>and one, the&nbsp;wave&nbsp;function must be&nbsp;continuous.&nbsp;Momentum&nbsp;being&nbsp;related to&nbsp;the&nbsp;space&nbsp;<br \/>derivatives&nbsp;of the&nbsp;wave&nbsp;function,&nbsp;the&nbsp;partial derivatives&nbsp;&part;&psi;\/&part;x,&nbsp;&part;&psi;\/&part;y&nbsp;and&nbsp;&part;&psi;\/&part;z<\/p>\n<p style=\"position:absolute;top:695px;left:768px;white-space:nowrap\" class=\"ft153\">&nbsp;<\/p>\n<p style=\"position:absolute;top:688px;left:773px;white-space:nowrap\" class=\"ft150\">must&nbsp;<\/p>\n<p style=\"position:absolute;top:719px;left:108px;white-space:nowrap\" class=\"ft157\">also&nbsp;be&nbsp;continuous&nbsp;and single&nbsp;valued everywhere.&nbsp;Thus, the&nbsp;important&nbsp;characteristics&nbsp;of wave&nbsp;<br \/>function are&nbsp;as&nbsp;follows:&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:799px;left:110px;white-space:nowrap\" class=\"ft150\">(1)&nbsp;<\/p>\n<p style=\"position:absolute;top:794px;left:164px;white-space:nowrap\" class=\"ft150\">&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:793px;left:187px;white-space:nowrap\" class=\"ft150\">must be&nbsp;finite, continuous&nbsp;and single&nbsp;valued everywhere.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:813px;left:108px;white-space:nowrap\" class=\"ft150\">&nbsp;<\/p>\n<p style=\"position:absolute;top:841px;left:110px;white-space:nowrap\" class=\"ft150\">(2)&nbsp;<\/p>\n<p style=\"position:absolute;top:834px;left:164px;white-space:nowrap\" class=\"ft150\">&part;&psi;\/&part;x,&nbsp;&nbsp;&part;&psi;\/&part;y&nbsp;&nbsp;and&nbsp;&nbsp;&part;&psi;\/&part;z&nbsp;&nbsp;must be&nbsp;finite, continuous&nbsp;and single&nbsp;valued&nbsp;<\/p>\n<p style=\"position:absolute;top:860px;left:110px;white-space:nowrap\" class=\"ft150\">everywhere.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:881px;left:108px;white-space:nowrap\" class=\"ft150\">&nbsp;<\/p>\n<p style=\"position:absolute;top:900px;left:110px;white-space:nowrap\" class=\"ft150\">(3)&nbsp;<\/p>\n<p style=\"position:absolute;top:894px;left:164px;white-space:nowrap\" class=\"ft150\">&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:894px;left:187px;white-space:nowrap\" class=\"ft150\">must&nbsp;be&nbsp;normalizable.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:910px;left:108px;white-space:nowrap\" class=\"ft159\">&nbsp;<br \/><b>Physical significance&nbsp;of&nbsp;wave&nbsp;function:&nbsp;&nbsp;<br \/><\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:967px;left:110px;white-space:nowrap\" class=\"ft1510\">We&nbsp;have&nbsp;already&nbsp;seen that the&nbsp;wave&nbsp;function has&nbsp;no&nbsp;direct&nbsp;physical&nbsp;significance.&nbsp;However, it&nbsp;<br \/>contains&nbsp;information about the&nbsp;system&nbsp;it represents&nbsp;and this&nbsp;can be&nbsp;extracted by&nbsp;appropriate&nbsp;<br \/>methods.&nbsp;Even though&nbsp;the&nbsp;wave&nbsp;function itself is&nbsp;not directly&nbsp;an observable&nbsp;quantity, the&nbsp;<br \/>square&nbsp;of the&nbsp;absolute&nbsp;value&nbsp;of the&nbsp;wave&nbsp;function is&nbsp;intimately&nbsp;related to&nbsp;the&nbsp;moving&nbsp;body&nbsp;and&nbsp;<br \/>is&nbsp;known&nbsp;as&nbsp;the&nbsp;probability&nbsp;density.&nbsp;This&nbsp;probability&nbsp;density&nbsp;is&nbsp;the&nbsp;quantum&nbsp;mechanical&nbsp;method&nbsp;<\/p>\n<\/div>\n<div id=\"page16-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224016.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft160\">15&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:110px;white-space:nowrap\" class=\"ft165\">of finding&nbsp;the&nbsp;body&nbsp;at a&nbsp;particular&nbsp;position at&nbsp;a particular time.&nbsp;The&nbsp;wave&nbsp;function&nbsp;carries&nbsp;<br \/>information about the&nbsp;particle&rsquo;s&nbsp;wave-like&nbsp;behaviour.&nbsp;It also&nbsp;provides&nbsp;information about the&nbsp;<br \/>momentum&nbsp;and energy&nbsp;of the&nbsp;particle&nbsp;at any&nbsp;instant of time.&nbsp;<\/p>\n<p style=\"position:absolute;top:143px;left:108px;white-space:nowrap\" class=\"ft161\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:165px;left:108px;white-space:nowrap\" class=\"ft161\"><b>Schrodinger&rsquo;s&nbsp;wave&nbsp;equation:&nbsp;[RGPV&nbsp;JUNE&nbsp;2013,&nbsp;DEC&nbsp;2013&nbsp;(7)]&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:182px;left:108px;white-space:nowrap\" class=\"ft166\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:233px;left:108px;white-space:nowrap\" class=\"ft162\">The&nbsp;motion&nbsp;of&nbsp;a&nbsp;free&nbsp;particle&nbsp;can&nbsp;be&nbsp;described&nbsp;<\/p>\n<p style=\"position:absolute;top:231px;left:406px;white-space:nowrap\" class=\"ft160\">by&nbsp;the&nbsp;wave&nbsp;equation.&nbsp;<\/p>\n<p style=\"position:absolute;top:247px;left:108px;white-space:nowrap\" class=\"ft167\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:269px;left:162px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;= A&nbsp;&nbsp;exp{-i(&omega;t&nbsp;&ndash;kx)}&nbsp;<\/p>\n<p style=\"position:absolute;top:276px;left:674px;white-space:nowrap\" class=\"ft160\">(1)<\/p>\n<p style=\"position:absolute;top:275px;left:707px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:324px;left:108px;white-space:nowrap\" class=\"ft160\">But&nbsp;<\/p>\n<p style=\"position:absolute;top:324px;left:162px;white-space:nowrap\" class=\"ft160\">&omega;&nbsp;= 2&nbsp;&nbsp;&pi;&nu;&nbsp;&nbsp;= 2&pi;&nbsp;<\/p>\n<p style=\"position:absolute;top:324px;left:322px;white-space:nowrap\" class=\"ft160\">(E\/h) = (E\/&#295;)&nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:348px;left:119px;white-space:nowrap\" class=\"ft160\">and&nbsp;k = 2&pi;\/&lambda;&nbsp;&nbsp;= 2&pi;&nbsp;(p\/h)&nbsp;= (p\/&#295;)&nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:388px;left:108px;white-space:nowrap\" class=\"ft160\">where&nbsp;&nbsp;E&nbsp;&nbsp;is&nbsp;<\/p>\n<p style=\"position:absolute;top:388px;left:237px;white-space:nowrap\" class=\"ft160\">the&nbsp;&nbsp;total&nbsp; energy&nbsp;&nbsp;and&nbsp;<\/p>\n<p style=\"position:absolute;top:388px;left:545px;white-space:nowrap\" class=\"ft160\">p&nbsp;&nbsp;is&nbsp;&nbsp;the&nbsp; momentum&nbsp; of&nbsp;<\/p>\n<p style=\"position:absolute;top:406px;left:108px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:426px;left:108px;white-space:nowrap\" class=\"ft168\">the&nbsp;particle.&nbsp;Substituting&nbsp;in the&nbsp;equation (1),&nbsp;we get,&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:481px;left:141px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;= A exp{-i&nbsp;(Et-px)}&nbsp;<\/p>\n<p style=\"position:absolute;top:488px;left:425px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:488px;left:676px;white-space:nowrap\" class=\"ft160\">(2)<\/p>\n<p style=\"position:absolute;top:487px;left:708px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:509px;left:108px;white-space:nowrap\" class=\"ft163\">&nbsp;<\/p>\n<p style=\"position:absolute;top:502px;left:260px;white-space:nowrap\" class=\"ft160\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:509px;left:425px;white-space:nowrap\" class=\"ft163\">&nbsp;<\/p>\n<p style=\"position:absolute;top:509px;left:590px;white-space:nowrap\" class=\"ft163\">&nbsp;<\/p>\n<p style=\"position:absolute;top:527px;left:108px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:544px;left:108px;white-space:nowrap\" class=\"ft160\">&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:593px;left:183px;white-space:nowrap\" class=\"ft160\">Differentiating&nbsp; equation&nbsp;&nbsp;(2)&nbsp;&nbsp;with&nbsp;<\/p>\n<p style=\"position:absolute;top:593px;left:590px;white-space:nowrap\" class=\"ft160\">respect&nbsp;&nbsp;to&nbsp;<\/p>\n<p style=\"position:absolute;top:624px;left:108px;white-space:nowrap\" class=\"ft160\">x twice,&nbsp;we&nbsp;get,&nbsp;<\/p>\n<p style=\"position:absolute;top:624px;left:425px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:624px;left:590px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:681px;left:141px;white-space:nowrap\" class=\"ft160\">&part;<\/p>\n<p style=\"position:absolute;top:679px;left:152px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:681px;left:161px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;=&nbsp;&nbsp;-p<\/p>\n<p style=\"position:absolute;top:679px;left:235px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:688px;left:244px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:681px;left:255px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;&nbsp;or p<\/p>\n<p style=\"position:absolute;top:679px;left:329px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:681px;left:338px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;=&nbsp;&#8211;&nbsp;<\/p>\n<p style=\"position:absolute;top:681px;left:401px;white-space:nowrap\" class=\"ft160\">&#295;<\/p>\n<p style=\"position:absolute;top:679px;left:412px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:687px;left:421px;white-space:nowrap\" class=\"ft160\">&nbsp;&nbsp;.&nbsp;<\/p>\n<p style=\"position:absolute;top:681px;left:451px;white-space:nowrap\" class=\"ft160\">&part;<\/p>\n<p style=\"position:absolute;top:679px;left:461px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:681px;left:471px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:690px;left:674px;white-space:nowrap\" class=\"ft160\">(3)<\/p>\n<p style=\"position:absolute;top:689px;left:707px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:708px;left:137px;white-space:nowrap\" class=\"ft160\">&part;x<\/p>\n<p style=\"position:absolute;top:706px;left:158px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:714px;left:168px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:708px;left:227px;white-space:nowrap\" class=\"ft160\">&#295;<\/p>\n<p style=\"position:absolute;top:706px;left:237px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:714px;left:247px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:708px;left:453px;white-space:nowrap\" class=\"ft160\">&part;x<\/p>\n<p style=\"position:absolute;top:706px;left:475px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:714px;left:484px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:715px;left:590px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:732px;left:108px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:744px;left:108px;white-space:nowrap\" class=\"ft160\">Differentiating&nbsp;equation&nbsp;(2)&nbsp;with respect to&nbsp;t, we&nbsp;get,&nbsp;<\/p>\n<p style=\"position:absolute;top:760px;left:108px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:784px;left:108px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:808px;left:119px;white-space:nowrap\" class=\"ft160\">&part;&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:815px;left:185px;white-space:nowrap\" class=\"ft160\">=<\/p>\n<p style=\"position:absolute;top:814px;left:195px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:815px;left:217px;white-space:nowrap\" class=\"ft160\">&#8211;&nbsp;iE<\/p>\n<p style=\"position:absolute;top:814px;left:261px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:815px;left:282px;white-space:nowrap\" class=\"ft160\">.&nbsp;<\/p>\n<p style=\"position:absolute;top:808px;left:304px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;&nbsp;or E&nbsp;&psi;&nbsp;&nbsp;=&nbsp;&#8211;&nbsp;<\/p>\n<p style=\"position:absolute;top:808px;left:469px;white-space:nowrap\" class=\"ft160\">&#295;&nbsp;.&nbsp;&part;&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:815px;left:667px;white-space:nowrap\" class=\"ft160\">(4)<\/p>\n<p style=\"position:absolute;top:814px;left:699px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:829px;left:119px;white-space:nowrap\" class=\"ft160\">&part;t&nbsp;<\/p>\n<p style=\"position:absolute;top:836px;left:174px;white-space:nowrap\" class=\"ft163\">&nbsp;<\/p>\n<p style=\"position:absolute;top:829px;left:249px;white-space:nowrap\" class=\"ft160\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:836px;left:264px;white-space:nowrap\" class=\"ft163\">&nbsp;<\/p>\n<p style=\"position:absolute;top:829px;left:476px;white-space:nowrap\" class=\"ft160\">i &nbsp; &part;t&nbsp;<\/p>\n<p style=\"position:absolute;top:836px;left:587px;white-space:nowrap\" class=\"ft163\">&nbsp;<\/p>\n<p style=\"position:absolute;top:869px;left:138px;white-space:nowrap\" class=\"ft160\">The&nbsp;total&nbsp;energy&nbsp;of the&nbsp;particle&nbsp;can be&nbsp;written as&nbsp;<\/p>\n<p style=\"position:absolute;top:943px;left:162px;white-space:nowrap\" class=\"ft160\">E<\/p>\n<p style=\"position:absolute;top:942px;left:172px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:943px;left:199px;white-space:nowrap\" class=\"ft160\">=<\/p>\n<p style=\"position:absolute;top:942px;left:210px;white-space:nowrap\" class=\"ft160\">&nbsp;&nbsp;p<\/p>\n<p style=\"position:absolute;top:932px;left:237px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:940px;left:247px;white-space:nowrap\" class=\"ft160\">&nbsp;&nbsp;+ U&nbsp;<\/p>\n<p style=\"position:absolute;top:943px;left:416px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:943px;left:667px;white-space:nowrap\" class=\"ft160\">(5)<\/p>\n<p style=\"position:absolute;top:942px;left:699px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:963px;left:119px;white-space:nowrap\" class=\"ft164\">&nbsp;<\/p>\n<p style=\"position:absolute;top:963px;left:174px;white-space:nowrap\" class=\"ft164\">&nbsp;<\/p>\n<p style=\"position:absolute;top:960px;left:216px;white-space:nowrap\" class=\"ft160\">2m<\/p>\n<p style=\"position:absolute;top:959px;left:238px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:963px;left:264px;white-space:nowrap\" class=\"ft164\">&nbsp;<\/p>\n<p style=\"position:absolute;top:963px;left:416px;white-space:nowrap\" class=\"ft164\">&nbsp;<\/p>\n<p style=\"position:absolute;top:963px;left:587px;white-space:nowrap\" class=\"ft164\">&nbsp;<\/p>\n<p style=\"position:absolute;top:974px;left:108px;white-space:nowrap\" class=\"ft169\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:1008px;left:108px;white-space:nowrap\" class=\"ft160\">Where&nbsp;U&nbsp;is&nbsp;the&nbsp;potential&nbsp;energy&nbsp;of the&nbsp;particle.&nbsp;Multiplying&nbsp;both sides&nbsp;of the&nbsp;equation by&nbsp;&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:1047px;left:108px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1074px;left:141px;white-space:nowrap\" class=\"ft160\">E&nbsp;<\/p>\n<p style=\"position:absolute;top:1067px;left:163px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp; &nbsp;= &nbsp;p<\/p>\n<p style=\"position:absolute;top:1065px;left:251px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:1067px;left:260px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;&nbsp;+ U&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:1076px;left:671px;white-space:nowrap\" class=\"ft160\">(6)<\/p>\n<p style=\"position:absolute;top:1075px;left:704px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1093px;left:237px;white-space:nowrap\" class=\"ft160\">2m<\/p>\n<p style=\"position:absolute;top:1092px;left:259px;white-space:nowrap\" class=\"ft160\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1096px;left:488px;white-space:nowrap\" class=\"ft164\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1101px;left:108px;white-space:nowrap\" class=\"ft1610\">&nbsp;<br \/>Substituting for E<\/p>\n<p style=\"position:absolute;top:1112px;left:302px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;and p<\/p>\n<p style=\"position:absolute;top:1109px;left:380px;white-space:nowrap\" class=\"ft162\">2<\/p>\n<p style=\"position:absolute;top:1112px;left:389px;white-space:nowrap\" class=\"ft160\">&psi;&nbsp;from&nbsp;equation (1.42) and (1.43)&nbsp;<\/p>\n<\/div>\n<div id=\"page17-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224017.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft170\">16&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft176\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:80px;left:129px;white-space:nowrap\" class=\"ft170\">&#8211;&nbsp;<\/p>\n<p style=\"position:absolute;top:73px;left:150px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;&nbsp;&part;&psi;&nbsp;&nbsp;= &nbsp;&#8211;&nbsp;&#295;<\/p>\n<p style=\"position:absolute;top:69px;left:300px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:77px;left:309px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:71px;left:323px;white-space:nowrap\" class=\"ft170\">&part;<\/p>\n<p style=\"position:absolute;top:69px;left:333px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:71px;left:343px;white-space:nowrap\" class=\"ft170\">&psi;&nbsp;&nbsp;+ U&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:80px;left:656px;white-space:nowrap\" class=\"ft170\">(7)<\/p>\n<p style=\"position:absolute;top:79px;left:689px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:107px;left:150px;white-space:nowrap\" class=\"ft170\">i<\/p>\n<p style=\"position:absolute;top:106px;left:161px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:100px;left:183px;white-space:nowrap\" class=\"ft170\">&part;t&nbsp;<\/p>\n<p style=\"position:absolute;top:107px;left:290px;white-space:nowrap\" class=\"ft170\">2m<\/p>\n<p style=\"position:absolute;top:106px;left:312px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:98px;left:324px;white-space:nowrap\" class=\"ft170\">&part;x<\/p>\n<p style=\"position:absolute;top:96px;left:346px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:104px;left:355px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:105px;left:522px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:126px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:141px;left:108px;white-space:nowrap\" class=\"ft170\">This is known&nbsp;as&nbsp;<\/p>\n<p style=\"position:absolute;top:141px;left:230px;white-space:nowrap\" class=\"ft172\"><b>Schrodinger&rsquo;s&nbsp;time&nbsp;dependent&nbsp;equation&nbsp;<\/b>in&nbsp;one&nbsp;dimension.&nbsp;<\/p>\n<p style=\"position:absolute;top:160px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:176px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:191px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:206px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:227px;left:108px;white-space:nowrap\" class=\"ft170\">The&nbsp;wave&nbsp;function&nbsp;&psi;&nbsp;in equation (2)&nbsp;may&nbsp;also&nbsp;be&nbsp;written&nbsp;as&nbsp;<\/p>\n<p style=\"position:absolute;top:239px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:259px;left:129px;white-space:nowrap\" class=\"ft170\">&psi;&nbsp;&nbsp;= A exp&nbsp;{-i&nbsp;(Et-px)} = A exp (-iEt). exp (ipx)&nbsp;<\/p>\n<p style=\"position:absolute;top:271px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:281px;left:270px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:281px;left:506px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:282px;left:626px;white-space:nowrap\" class=\"ft173\">&#295;<\/p>\n<p style=\"position:absolute;top:287px;left:636px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:314px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:333px;left:129px;white-space:nowrap\" class=\"ft170\">&psi;&nbsp;= &nbsp;&Phi;&nbsp;exp (-iEt)&nbsp;<\/p>\n<p style=\"position:absolute;top:338px;left:641px;white-space:nowrap\" class=\"ft170\">(8 )&nbsp;<\/p>\n<p style=\"position:absolute;top:352px;left:281px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:386px;left:108px;white-space:nowrap\" class=\"ft177\">&nbsp;<br \/>where&nbsp;&Phi;&nbsp;is&nbsp;a&nbsp;position&nbsp;dependent function.&nbsp;Substituting&nbsp;this&nbsp;form&nbsp;of&nbsp;&psi;&nbsp;in equation&nbsp;(6),&nbsp;<\/p>\n<p style=\"position:absolute;top:424px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:439px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:454px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:488px;left:162px;white-space:nowrap\" class=\"ft170\">E<\/p>\n<p style=\"position:absolute;top:481px;left:173px;white-space:nowrap\" class=\"ft170\">&Phi;&nbsp;exp(-iEt)&nbsp;&nbsp;= &nbsp;p<\/p>\n<p style=\"position:absolute;top:477px;left:363px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:485px;left:372px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:481px;left:378px;white-space:nowrap\" class=\"ft170\">&Phi;&nbsp;exp(-iEt)&nbsp;+&nbsp;U&Phi;&nbsp;exp(-iEt)&nbsp;<\/p>\n<p style=\"position:absolute;top:498px;left:259px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:505px;left:350px;white-space:nowrap\" class=\"ft170\">2m<\/p>\n<p style=\"position:absolute;top:504px;left:372px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:498px;left:465px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:498px;left:637px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:518px;left:108px;white-space:nowrap\" class=\"ft178\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:567px;left:108px;white-space:nowrap\" class=\"ft173\">or E<\/p>\n<p style=\"position:absolute;top:561px;left:149px;white-space:nowrap\" class=\"ft173\">&Phi;&nbsp;exp(-iEt) =&nbsp;&#8211;&nbsp;<\/p>\n<p style=\"position:absolute;top:560px;left:318px;white-space:nowrap\" class=\"ft173\">&#295;<\/p>\n<p style=\"position:absolute;top:559px;left:328px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:567px;left:337px;white-space:nowrap\" class=\"ft173\">&nbsp;&nbsp;.&nbsp;<\/p>\n<p style=\"position:absolute;top:560px;left:379px;white-space:nowrap\" class=\"ft173\">&part;<\/p>\n<p style=\"position:absolute;top:559px;left:389px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:561px;left:398px;white-space:nowrap\" class=\"ft173\">&Phi;&nbsp;. exp(-iEt) + U&Phi;&nbsp;exp(-iEt)<\/p>\n<p style=\"position:absolute;top:566px;left:693px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:584px;left:237px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:584px;left:322px;white-space:nowrap\" class=\"ft170\">2m &nbsp; &part;x<\/p>\n<p style=\"position:absolute;top:582px;left:398px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:591px;left:408px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:584px;left:502px;white-space:nowrap\" class=\"ft170\">&#295;&nbsp;<\/p>\n<p style=\"position:absolute;top:585px;left:664px;white-space:nowrap\" class=\"ft173\">&#295;<\/p>\n<p style=\"position:absolute;top:591px;left:675px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:604px;left:108px;white-space:nowrap\" class=\"ft179\">&nbsp;<br \/>or<\/p>\n<p style=\"position:absolute;top:626px;left:130px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:620px;left:150px;white-space:nowrap\" class=\"ft170\">&part;<\/p>\n<p style=\"position:absolute;top:618px;left:161px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:620px;left:170px;white-space:nowrap\" class=\"ft170\">&Phi;&nbsp;exp(-iEt) &nbsp;+&nbsp;2m&nbsp;(E-U)&Phi;&nbsp;exp(-iEt) &nbsp;= &nbsp;0&nbsp;<\/p>\n<p style=\"position:absolute;top:645px;left:151px;white-space:nowrap\" class=\"ft170\">&part;x<\/p>\n<p style=\"position:absolute;top:643px;left:173px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:651px;left:183px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:645px;left:266px;white-space:nowrap\" class=\"ft170\">&#295; &nbsp; &nbsp; &nbsp;&#295;<\/p>\n<p style=\"position:absolute;top:643px;left:352px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:651px;left:361px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:646px;left:510px;white-space:nowrap\" class=\"ft173\">&#295;<\/p>\n<p style=\"position:absolute;top:651px;left:520px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:680px;left:110px;white-space:nowrap\" class=\"ft170\">or &nbsp;<\/p>\n<p style=\"position:absolute;top:673px;left:153px;white-space:nowrap\" class=\"ft170\">&part;<\/p>\n<p style=\"position:absolute;top:670px;left:164px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:673px;left:173px;white-space:nowrap\" class=\"ft170\">&psi;&nbsp;&nbsp;+ &nbsp;2m&nbsp;(E-U)&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:682px;left:377px;white-space:nowrap\" class=\"ft170\">= &nbsp;0<\/p>\n<p style=\"position:absolute;top:681px;left:420px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:682px;left:671px;white-space:nowrap\" class=\"ft170\">(9)<\/p>\n<p style=\"position:absolute;top:681px;left:703px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:696px;left:153px;white-space:nowrap\" class=\"ft171\">&part;x<\/p>\n<p style=\"position:absolute;top:695px;left:171px;white-space:nowrap\" class=\"ft174\">2<\/p>\n<p style=\"position:absolute;top:700px;left:179px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:696px;left:246px;white-space:nowrap\" class=\"ft171\">&#295;<\/p>\n<p style=\"position:absolute;top:695px;left:255px;white-space:nowrap\" class=\"ft174\">2<\/p>\n<p style=\"position:absolute;top:700px;left:263px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:702px;left:353px;white-space:nowrap\" class=\"ft175\">&nbsp;<\/p>\n<p style=\"position:absolute;top:702px;left:525px;white-space:nowrap\" class=\"ft175\">&nbsp;<\/p>\n<p style=\"position:absolute;top:706px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:724px;left:110px;white-space:nowrap\" class=\"ft1710\">&nbsp;<br \/>This&nbsp;is&nbsp;the&nbsp;Schrodinger&rsquo;s&nbsp;wave&nbsp;equation in one&nbsp;dimension.&nbsp;In three&nbsp;dimensions, the&nbsp;above&nbsp;<br \/>equation may&nbsp;be&nbsp;written&nbsp;as&nbsp;<\/p>\n<p style=\"position:absolute;top:798px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:813px;left:110px;white-space:nowrap\" class=\"ft170\">&part;<\/p>\n<p style=\"position:absolute;top:811px;left:120px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:813px;left:130px;white-space:nowrap\" class=\"ft170\">&psi;&nbsp;+&nbsp;<\/p>\n<p style=\"position:absolute;top:813px;left:183px;white-space:nowrap\" class=\"ft170\">&part;<\/p>\n<p style=\"position:absolute;top:811px;left:194px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:813px;left:203px;white-space:nowrap\" class=\"ft170\">&psi;&nbsp;+&nbsp;<\/p>\n<p style=\"position:absolute;top:813px;left:257px;white-space:nowrap\" class=\"ft170\">&part;<\/p>\n<p style=\"position:absolute;top:811px;left:267px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:813px;left:276px;white-space:nowrap\" class=\"ft170\">&psi;&nbsp;+&nbsp;2m(E-U)&psi;&nbsp;= 0&nbsp;<\/p>\n<p style=\"position:absolute;top:840px;left:110px;white-space:nowrap\" class=\"ft170\">&part;x<\/p>\n<p style=\"position:absolute;top:838px;left:131px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:846px;left:141px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:840px;left:182px;white-space:nowrap\" class=\"ft170\">&part;y<\/p>\n<p style=\"position:absolute;top:838px;left:203px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:846px;left:213px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:840px;left:254px;white-space:nowrap\" class=\"ft170\">&part;z<\/p>\n<p style=\"position:absolute;top:838px;left:275px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:846px;left:285px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:840px;left:325px;white-space:nowrap\" class=\"ft170\">&#295;<\/p>\n<p style=\"position:absolute;top:838px;left:336px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:846px;left:346px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:867px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:891px;left:110px;white-space:nowrap\" class=\"ft170\">or<\/p>\n<p style=\"position:absolute;top:890px;left:131px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:885px;left:151px;white-space:nowrap\" class=\"ft170\">&nabla;<\/p>\n<p style=\"position:absolute;top:882px;left:164px;white-space:nowrap\" class=\"ft171\">2<\/p>\n<p style=\"position:absolute;top:885px;left:174px;white-space:nowrap\" class=\"ft170\">&psi;&nbsp;+&nbsp;2m(E-U)&psi;&nbsp;=0&nbsp;<\/p>\n<p style=\"position:absolute;top:907px;left:228px;white-space:nowrap\" class=\"ft173\">&#295;<\/p>\n<p style=\"position:absolute;top:906px;left:238px;white-space:nowrap\" class=\"ft175\">2<\/p>\n<p style=\"position:absolute;top:912px;left:247px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:920px;left:108px;white-space:nowrap\" class=\"ft170\">&nbsp;<\/p>\n<p style=\"position:absolute;top:932px;left:110px;white-space:nowrap\" class=\"ft170\">This&nbsp;equation is&nbsp;known&nbsp;as&nbsp;<\/p>\n<p style=\"position:absolute;top:931px;left:303px;white-space:nowrap\" class=\"ft172\"><b>the steady&nbsp;state or&nbsp;time<\/b>&nbsp;<b>independent Schrodinger&nbsp;wave&nbsp;equation&nbsp;<\/b>in&nbsp;<\/p>\n<p style=\"position:absolute;top:962px;left:110px;white-space:nowrap\" class=\"ft170\">three<\/p>\n<p style=\"position:absolute;top:961px;left:149px;white-space:nowrap\" class=\"ft172\"><b>&nbsp;<\/b>dimensions.&nbsp;<\/p>\n<p style=\"position:absolute;top:992px;left:110px;white-space:nowrap\" class=\"ft1711\">&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<\/div>\n<div id=\"page18-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224018.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft180\">17&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft181\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:69px;left:116px;white-space:nowrap\" class=\"ft180\">S.NO&nbsp;<\/p>\n<p style=\"position:absolute;top:69px;left:280px;white-space:nowrap\" class=\"ft180\">RGPV QUESTION&nbsp;<\/p>\n<p style=\"position:absolute;top:69px;left:545px;white-space:nowrap\" class=\"ft180\">YEAR&nbsp;<\/p>\n<p style=\"position:absolute;top:69px;left:695px;white-space:nowrap\" class=\"ft180\">MARKS&nbsp;<\/p>\n<p style=\"position:absolute;top:92px;left:116px;white-space:nowrap\" class=\"ft180\">Q.1&nbsp;<\/p>\n<p style=\"position:absolute;top:92px;left:197px;white-space:nowrap\" class=\"ft180\">Discuss&nbsp;the&nbsp;concept of&nbsp;wave&nbsp;function&nbsp;<\/p>\n<p style=\"position:absolute;top:113px;left:197px;white-space:nowrap\" class=\"ft180\">associated with&nbsp;the&nbsp;particle.&nbsp;Give&nbsp;<\/p>\n<p style=\"position:absolute;top:135px;left:197px;white-space:nowrap\" class=\"ft180\">examples&nbsp;of&nbsp;admissible&nbsp;wave&nbsp;function.&nbsp;<\/p>\n<p style=\"position:absolute;top:157px;left:197px;white-space:nowrap\" class=\"ft180\">Why&nbsp;derivatives&nbsp;of&nbsp;wave&nbsp;function&nbsp;<\/p>\n<p style=\"position:absolute;top:179px;left:197px;white-space:nowrap\" class=\"ft180\">should be&nbsp;continuous&nbsp;everywhere?&nbsp;<\/p>\n<p style=\"position:absolute;top:92px;left:526px;white-space:nowrap\" class=\"ft180\">JUNE2013&nbsp;<\/p>\n<p style=\"position:absolute;top:92px;left:717px;white-space:nowrap\" class=\"ft180\">7&nbsp;<\/p>\n<p style=\"position:absolute;top:202px;left:116px;white-space:nowrap\" class=\"ft180\">Q.2&nbsp;<\/p>\n<p style=\"position:absolute;top:202px;left:197px;white-space:nowrap\" class=\"ft180\">&nbsp;Derive&nbsp;&nbsp;Schrodinger&rsquo;s&nbsp;time&nbsp;dependent&nbsp;<\/p>\n<p style=\"position:absolute;top:224px;left:197px;white-space:nowrap\" class=\"ft180\">equation&nbsp;for matter wave?&nbsp;<\/p>\n<p style=\"position:absolute;top:202px;left:528px;white-space:nowrap\" class=\"ft180\">DEC&nbsp;2013&nbsp;<\/p>\n<p style=\"position:absolute;top:202px;left:717px;white-space:nowrap\" class=\"ft180\">7&nbsp;<\/p>\n<p style=\"position:absolute;top:247px;left:108px;white-space:nowrap\" class=\"ft180\">&nbsp;<\/p>\n<p style=\"position:absolute;top:269px;left:108px;white-space:nowrap\" class=\"ft180\">&nbsp;<\/p>\n<p style=\"position:absolute;top:291px;left:108px;white-space:nowrap\" class=\"ft180\">&nbsp;<\/p>\n<p style=\"position:absolute;top:313px;left:108px;white-space:nowrap\" class=\"ft180\">&nbsp;<\/p>\n<p style=\"position:absolute;top:336px;left:108px;white-space:nowrap\" class=\"ft180\">&nbsp;<\/p>\n<p style=\"position:absolute;top:357px;left:108px;white-space:nowrap\" class=\"ft180\">&nbsp;<\/p>\n<p style=\"position:absolute;top:379px;left:108px;white-space:nowrap\" class=\"ft180\">&nbsp;<\/p>\n<p style=\"position:absolute;top:379px;left:324px;white-space:nowrap\" class=\"ft180\">&nbsp;<\/p>\n<\/div>\n<div id=\"page19-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224019.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft190\">18&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:46px;left:379px;white-space:nowrap\" class=\"ft191\"><b>UNIT&nbsp;1\/LECTURE&nbsp;7&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:72px;left:104px;white-space:nowrap\" class=\"ft192\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:94px;left:108px;white-space:nowrap\" class=\"ft192\"><b>APPLICATIONS&nbsp;OF&nbsp;SCHRODINGER&rsquo;S&nbsp;EQUATION:&nbsp;[RGPV&nbsp;Dec2013&nbsp;(7)]&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:107px;left:108px;white-space:nowrap\" class=\"ft192\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:126px;left:108px;white-space:nowrap\" class=\"ft192\"><b>Case&nbsp;of a&nbsp;free&nbsp;particle:&nbsp;&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:144px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:165px;left:164px;white-space:nowrap\" class=\"ft190\">A&nbsp;free&nbsp;particle&nbsp;is&nbsp;defined as&nbsp;one&nbsp;which is&nbsp;not acted upon by&nbsp;any&nbsp;external&nbsp;force&nbsp;that&nbsp;<\/p>\n<p style=\"position:absolute;top:195px;left:110px;white-space:nowrap\" class=\"ft190\">modifies&nbsp;its&nbsp;motion.&nbsp;Hence, the&nbsp;potential&nbsp;energy&nbsp;U&nbsp;in the&nbsp;<\/p>\n<p style=\"position:absolute;top:195px;left:549px;white-space:nowrap\" class=\"ft193\">Schrodinger&rsquo;s&nbsp;equation is&nbsp;a&nbsp;constant&nbsp;<\/p>\n<p style=\"position:absolute;top:225px;left:110px;white-space:nowrap\" class=\"ft193\">and does&nbsp;not&nbsp;<\/p>\n<p style=\"position:absolute;top:224px;left:219px;white-space:nowrap\" class=\"ft190\">depend on position&nbsp;or&nbsp;time.&nbsp;For&nbsp;convenience, the&nbsp;potential&nbsp;energy may be&nbsp;<\/p>\n<p style=\"position:absolute;top:254px;left:110px;white-space:nowrap\" class=\"ft190\">assumed to&nbsp;be&nbsp;zero.&nbsp;Then, the&nbsp;Schrodinger&rsquo;s&nbsp;equation for&nbsp;the&nbsp;particle&nbsp;becomes&nbsp;<\/p>\n<p style=\"position:absolute;top:279px;left:108px;white-space:nowrap\" class=\"ft198\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:329px;left:173px;white-space:nowrap\" class=\"ft190\">&part;<\/p>\n<p style=\"position:absolute;top:327px;left:183px;white-space:nowrap\" class=\"ft194\">2<\/p>\n<p style=\"position:absolute;top:329px;left:193px;white-space:nowrap\" class=\"ft190\">&psi;&nbsp;&nbsp;+&nbsp;2m&nbsp;E&psi;&nbsp;<\/p>\n<p style=\"position:absolute;top:338px;left:344px;white-space:nowrap\" class=\"ft190\">= &nbsp;0<\/p>\n<p style=\"position:absolute;top:337px;left:387px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:338px;left:659px;white-space:nowrap\" class=\"ft190\">(10)<\/p>\n<p style=\"position:absolute;top:337px;left:702px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:353px;left:173px;white-space:nowrap\" class=\"ft194\">&part;x<\/p>\n<p style=\"position:absolute;top:352px;left:191px;white-space:nowrap\" class=\"ft195\">2<\/p>\n<p style=\"position:absolute;top:356px;left:198px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:353px;left:257px;white-space:nowrap\" class=\"ft194\">&#295;<\/p>\n<p style=\"position:absolute;top:352px;left:266px;white-space:nowrap\" class=\"ft195\">2<\/p>\n<p style=\"position:absolute;top:356px;left:273px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:359px;left:318px;white-space:nowrap\" class=\"ft196\">&nbsp;<\/p>\n<p style=\"position:absolute;top:359px;left:507px;white-space:nowrap\" class=\"ft196\">&nbsp;<\/p>\n<p style=\"position:absolute;top:363px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:374px;left:108px;white-space:nowrap\" class=\"ft190\">Where&nbsp;E&nbsp;is&nbsp;the&nbsp;total&nbsp;energy&nbsp;of the&nbsp;particle&nbsp;which is&nbsp;purely&nbsp;kinetic.&nbsp;This&nbsp;is&nbsp;of the&nbsp;form,&nbsp;<\/p>\n<p style=\"position:absolute;top:394px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:409px;left:173px;white-space:nowrap\" class=\"ft190\">&part;<\/p>\n<p style=\"position:absolute;top:407px;left:183px;white-space:nowrap\" class=\"ft194\">2<\/p>\n<p style=\"position:absolute;top:409px;left:193px;white-space:nowrap\" class=\"ft190\">&psi;&nbsp;+ k<\/p>\n<p style=\"position:absolute;top:407px;left:248px;white-space:nowrap\" class=\"ft194\">2<\/p>\n<p style=\"position:absolute;top:409px;left:258px;white-space:nowrap\" class=\"ft190\">&psi;&nbsp; &nbsp;= 0&nbsp;<\/p>\n<p style=\"position:absolute;top:431px;left:173px;white-space:nowrap\" class=\"ft193\">&part;x<\/p>\n<p style=\"position:absolute;top:430px;left:193px;white-space:nowrap\" class=\"ft194\">2<\/p>\n<p style=\"position:absolute;top:437px;left:202px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:445px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:461px;left:108px;white-space:nowrap\" class=\"ft190\">Where&nbsp;k<\/p>\n<p style=\"position:absolute;top:457px;left:171px;white-space:nowrap\" class=\"ft194\">2<\/p>\n<p style=\"position:absolute;top:467px;left:179px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:460px;left:183px;white-space:nowrap\" class=\"ft190\">=&nbsp;2mE\/&#295;<\/p>\n<p style=\"position:absolute;top:457px;left:246px;white-space:nowrap\" class=\"ft194\">2<\/p>\n<p style=\"position:absolute;top:467px;left:254px;white-space:nowrap\" class=\"ft190\">.&nbsp;<\/p>\n<p style=\"position:absolute;top:461px;left:263px;white-space:nowrap\" class=\"ft190\">The&nbsp;solution&nbsp;of&nbsp;this&nbsp;equation may&nbsp;be&nbsp;written as&nbsp;<\/p>\n<p style=\"position:absolute;top:480px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:495px;left:183px;white-space:nowrap\" class=\"ft190\">&psi;&nbsp;= A cos kx + B sin kx&nbsp;<\/p>\n<p style=\"position:absolute;top:518px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:543px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:561px;left:108px;white-space:nowrap\" class=\"ft197\">Solving<\/p>\n<p style=\"position:absolute;top:560px;left:156px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:561px;left:162px;white-space:nowrap\" class=\"ft197\">for&nbsp; &nbsp;the&nbsp;&nbsp; constants&nbsp;&nbsp;&nbsp;A&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;&nbsp;B&nbsp;&nbsp;&nbsp;pose&nbsp;&nbsp; some&nbsp;<\/p>\n<p style=\"position:absolute;top:560px;left:506px;white-space:nowrap\" class=\"ft190\">difficulties&nbsp;because&nbsp;we&nbsp;cannot apply&nbsp;any&nbsp;<\/p>\n<p style=\"position:absolute;top:582px;left:108px;white-space:nowrap\" class=\"ft190\">boundary&nbsp;conditions&nbsp;on the&nbsp;wave&nbsp;function as&nbsp;it represents&nbsp;a&nbsp;single&nbsp;wave&nbsp;which is&nbsp;not&nbsp;localized&nbsp;<\/p>\n<p style=\"position:absolute;top:604px;left:108px;white-space:nowrap\" class=\"ft190\">and not&nbsp;normalizable.&nbsp;Since&nbsp;the&nbsp;solution has&nbsp;not&nbsp;imposed any&nbsp;restriction&nbsp;on&nbsp;the&nbsp;value&nbsp;of k, the&nbsp;<\/p>\n<p style=\"position:absolute;top:626px;left:108px;white-space:nowrap\" class=\"ft190\">free&nbsp;particle&nbsp;is&nbsp;permitted to&nbsp;have&nbsp;any&nbsp;value&nbsp;of energy&nbsp;given by&nbsp;the&nbsp;equation,&nbsp;<\/p>\n<p style=\"position:absolute;top:640px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:654px;left:183px;white-space:nowrap\" class=\"ft190\">E = &#295;<\/p>\n<p style=\"position:absolute;top:652px;left:237px;white-space:nowrap\" class=\"ft194\">2<\/p>\n<p style=\"position:absolute;top:662px;left:246px;white-space:nowrap\" class=\"ft190\">k<\/p>\n<p style=\"position:absolute;top:652px;left:257px;white-space:nowrap\" class=\"ft194\">2<\/p>\n<p style=\"position:absolute;top:662px;left:267px;white-space:nowrap\" class=\"ft190\">\/2m<\/p>\n<p style=\"position:absolute;top:661px;left:299px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:673px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:687px;left:108px;white-space:nowrap\" class=\"ft199\">Since&nbsp;the&nbsp;total&nbsp;energy&nbsp;is&nbsp;purely&nbsp;kinetic, the&nbsp;momentum&nbsp;of the&nbsp;particle&nbsp;would be&nbsp;p =&nbsp;&#295;k&nbsp;or&nbsp;h\/&lambda;.&nbsp;<br \/>This&nbsp;is&nbsp;just&nbsp;what we&nbsp;would expect,&nbsp;since&nbsp;we&nbsp;have&nbsp;constructed the&nbsp;Schrodinger&nbsp;equation to&nbsp;yield&nbsp;<br \/>the&nbsp;solution&nbsp;for&nbsp;the&nbsp;free&nbsp;particle&nbsp;corresponding&nbsp;to&nbsp;a&nbsp;de&nbsp;Broglie&nbsp;wave.&nbsp;<\/p>\n<p style=\"position:absolute;top:786px;left:108px;white-space:nowrap\" class=\"ft192\"><b>&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:801px;left:108px;white-space:nowrap\" class=\"ft192\"><b>Particle&nbsp;in a&nbsp;one&nbsp;dimensional&nbsp;potential&nbsp;box:<\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:824px;left:108px;white-space:nowrap\" class=\"ft193\">The&nbsp;simplest&nbsp;problem&nbsp;for which&nbsp;Schrodinger&rsquo;s&nbsp;<\/p>\n<p style=\"position:absolute;top:823px;left:449px;white-space:nowrap\" class=\"ft190\">time&nbsp;independent equation can be&nbsp;applied and&nbsp;<\/p>\n<p style=\"position:absolute;top:845px;left:108px;white-space:nowrap\" class=\"ft190\">solved is&nbsp;the&nbsp;case&nbsp;of a&nbsp;particle&nbsp;trapped in a&nbsp;box&nbsp;with impenetrable&nbsp;walls.&nbsp;<\/p>\n<p style=\"position:absolute;top:858px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:876px;left:250px;white-space:nowrap\" class=\"ft190\">Consider a particle&nbsp;of&nbsp;mass&nbsp;m and&nbsp;energy&nbsp;E&nbsp;travelling&nbsp;along&nbsp;x-axis&nbsp;inside&nbsp;a&nbsp;<\/p>\n<p style=\"position:absolute;top:907px;left:110px;white-space:nowrap\" class=\"ft190\">box&nbsp;of width&nbsp;L.&nbsp;The&nbsp;particle&nbsp;is&nbsp;thus&nbsp;restricted to&nbsp;move&nbsp;inside&nbsp;the&nbsp;box&nbsp;by&nbsp;reflections&nbsp;at x=0&nbsp;and&nbsp;<\/p>\n<p style=\"position:absolute;top:944px;left:110px;white-space:nowrap\" class=\"ft190\">x=L&nbsp;(Fig. 1).&nbsp;<\/p>\n<p style=\"position:absolute;top:967px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:989px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1011px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1033px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1055px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1077px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1099px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1121px;left:108px;white-space:nowrap\" class=\"ft190\">&nbsp;<\/p>\n<\/div>\n<div id=\"page20-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224020.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft200\">19&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft205\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:68px;left:108px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:90px;left:108px;white-space:nowrap\" class=\"ft200\">The&nbsp;particle&nbsp;does&nbsp;not lose&nbsp;any&nbsp;energy&nbsp;when it&nbsp;collides&nbsp;with the&nbsp;walls&nbsp;and hence&nbsp;the&nbsp;total&nbsp;energy&nbsp;<\/p>\n<p style=\"position:absolute;top:112px;left:108px;white-space:nowrap\" class=\"ft200\">of the&nbsp;particle&nbsp;remains&nbsp;constant.&nbsp;The&nbsp;potential&nbsp;energy&nbsp;of the&nbsp;particle&nbsp;is&nbsp;considered&nbsp;to&nbsp;be&nbsp;zero&nbsp;<\/p>\n<p style=\"position:absolute;top:134px;left:108px;white-space:nowrap\" class=\"ft200\">inside&nbsp;the&nbsp;box&nbsp;and<\/p>\n<p style=\"position:absolute;top:136px;left:246px;white-space:nowrap\" class=\"ft201\">&nbsp;<\/p>\n<p style=\"position:absolute;top:134px;left:251px;white-space:nowrap\" class=\"ft200\">Infinite&nbsp;outside.&nbsp;<\/p>\n<p style=\"position:absolute;top:135px;left:372px;white-space:nowrap\" class=\"ft202\">Since the total&nbsp;energy&nbsp;of the&nbsp;<\/p>\n<p style=\"position:absolute;top:134px;left:582px;white-space:nowrap\" class=\"ft200\">particle&nbsp;cannot be&nbsp;infinite, it is&nbsp;<\/p>\n<p style=\"position:absolute;top:156px;left:108px;white-space:nowrap\" class=\"ft200\">restricted to&nbsp;move&nbsp;within the&nbsp;box.&nbsp;The&nbsp;example&nbsp;is&nbsp;an oversimplified case&nbsp;of an electron acted&nbsp;<\/p>\n<p style=\"position:absolute;top:178px;left:108px;white-space:nowrap\" class=\"ft200\">upon by&nbsp;the&nbsp;electrostatic&nbsp;potential of&nbsp;the&nbsp;ion&nbsp;cores&nbsp;in&nbsp;a crystal lattice.&nbsp;Since&nbsp;the&nbsp;particle&nbsp;cannot&nbsp;<\/p>\n<p style=\"position:absolute;top:200px;left:108px;white-space:nowrap\" class=\"ft200\">exist outside&nbsp;the&nbsp;box,&nbsp;<\/p>\n<p style=\"position:absolute;top:227px;left:120px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:251px;left:162px;white-space:nowrap\" class=\"ft200\">&psi;&nbsp;=&nbsp;0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for&nbsp; &nbsp; &nbsp; x &le;&nbsp;0 &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;x &ge;&nbsp;L&nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:257px;left:540px;white-space:nowrap\" class=\"ft200\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:258px;left:648px;white-space:nowrap\" class=\"ft200\">(1)<\/p>\n<p style=\"position:absolute;top:257px;left:680px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:271px;left:108px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:283px;left:108px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:296px;left:108px;white-space:nowrap\" class=\"ft203\">We&nbsp;have&nbsp;to&nbsp;evaluate&nbsp;the&nbsp;wave&nbsp;function&nbsp;inside&nbsp;the&nbsp;box.&nbsp;<\/p>\n<p style=\"position:absolute;top:315px;left:108px;white-space:nowrap\" class=\"ft200\">The&nbsp;Schrodinger&rsquo;s&nbsp;equation&nbsp;(1.48) becomes&nbsp;<\/p>\n<p style=\"position:absolute;top:352px;left:108px;white-space:nowrap\" class=\"ft200\">&part;<\/p>\n<p style=\"position:absolute;top:350px;left:119px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:352px;left:128px;white-space:nowrap\" class=\"ft200\">&psi;&nbsp;&nbsp;+ &nbsp;2m&nbsp;&nbsp;E&psi;&nbsp; &nbsp;=&nbsp;&nbsp;0 &nbsp;for &nbsp;0&nbsp;&lt;&nbsp;x&nbsp;&lt;&nbsp;L&nbsp;<\/p>\n<p style=\"position:absolute;top:361px;left:671px;white-space:nowrap\" class=\"ft200\">(2)<\/p>\n<p style=\"position:absolute;top:360px;left:704px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:379px;left:108px;white-space:nowrap\" class=\"ft200\">&part;x<\/p>\n<p style=\"position:absolute;top:377px;left:130px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:385px;left:139px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:379px;left:198px;white-space:nowrap\" class=\"ft200\">&#295;<\/p>\n<p style=\"position:absolute;top:377px;left:209px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:385px;left:218px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:386px;left:320px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:386px;left:599px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:407px;left:108px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:439px;left:108px;white-space:nowrap\" class=\"ft200\">&psi;&nbsp;&nbsp;=&nbsp;&nbsp;A sin&nbsp;(2mE)<\/p>\n<p style=\"position:absolute;top:435px;left:289px;white-space:nowrap\" class=\"ft203\">1\/2<\/p>\n<p style=\"position:absolute;top:443px;left:317px;white-space:nowrap\" class=\"ft200\">&nbsp;&nbsp;x + B cos (2mE&nbsp;)<\/p>\n<p style=\"position:absolute;top:435px;left:498px;white-space:nowrap\" class=\"ft203\">1\/2<\/p>\n<p style=\"position:absolute;top:444px;left:527px;white-space:nowrap\" class=\"ft200\">&nbsp; &nbsp;x<\/p>\n<p style=\"position:absolute;top:443px;left:570px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:446px;left:671px;white-space:nowrap\" class=\"ft200\">(3)<\/p>\n<p style=\"position:absolute;top:445px;left:704px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:471px;left:108px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:464px;left:259px;white-space:nowrap\" class=\"ft200\">&#295;<\/p>\n<p style=\"position:absolute;top:462px;left:270px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:470px;left:280px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:464px;left:461px;white-space:nowrap\" class=\"ft200\">&#295;<\/p>\n<p style=\"position:absolute;top:462px;left:471px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:470px;left:481px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:471px;left:599px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:484px;left:108px;white-space:nowrap\" class=\"ft206\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:517px;left:108px;white-space:nowrap\" class=\"ft207\">where&nbsp;A&nbsp;and B&nbsp;are&nbsp;constants.&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:557px;left:108px;white-space:nowrap\" class=\"ft209\">Applying&nbsp;the&nbsp;boundary&nbsp;condition that&nbsp;&psi;=0 at&nbsp;x&nbsp;=&nbsp;0,&nbsp;equation&nbsp;3&nbsp;becomes&nbsp;<br \/>&nbsp;<br \/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;A&nbsp;sin 0 +&nbsp;B&nbsp;cos&nbsp;0 =&nbsp;0 or&nbsp;B&nbsp;=&nbsp;0.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:626px;left:108px;white-space:nowrap\" class=\"ft2011\">&nbsp;<br \/>Again,&nbsp;we&nbsp;have&nbsp;&psi;&nbsp;=&nbsp;0 at&nbsp;x =&nbsp;L.&nbsp;Then,&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:715px;left:124px;white-space:nowrap\" class=\"ft200\">A.sin(2mE)<\/p>\n<p style=\"position:absolute;top:705px;left:209px;white-space:nowrap\" class=\"ft203\">1\/2<\/p>\n<p style=\"position:absolute;top:715px;left:229px;white-space:nowrap\" class=\"ft200\">.L=0&nbsp;<\/p>\n<p style=\"position:absolute;top:721px;left:108px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:737px;left:200px;white-space:nowrap\" class=\"ft200\">&#295;<\/p>\n<p style=\"position:absolute;top:734px;left:209px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:743px;left:216px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:759px;left:119px;white-space:nowrap\" class=\"ft200\">If A&nbsp;=&nbsp;0, the&nbsp;wave&nbsp;function will&nbsp;become&nbsp;zero&nbsp;irrespective&nbsp;of the&nbsp;value&nbsp;of x.&nbsp;Hence, A&nbsp;cannot be&nbsp;<\/p>\n<p style=\"position:absolute;top:786px;left:108px;white-space:nowrap\" class=\"ft200\">zero.&nbsp;<\/p>\n<p style=\"position:absolute;top:806px;left:108px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:821px;left:114px;white-space:nowrap\" class=\"ft200\">Therefore, sin(2mE)<\/p>\n<p style=\"position:absolute;top:819px;left:294px;white-space:nowrap\" class=\"ft203\">1\/2<\/p>\n<p style=\"position:absolute;top:828px;left:322px;white-space:nowrap\" class=\"ft200\">.L=0<\/p>\n<p style=\"position:absolute;top:827px;left:365px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:858px;left:129px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:849px;left:286px;white-space:nowrap\" class=\"ft200\">&#295;<\/p>\n<p style=\"position:absolute;top:847px;left:297px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:855px;left:307px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:858px;left:561px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:896px;left:129px;white-space:nowrap\" class=\"ft200\">or (2mE)<\/p>\n<p style=\"position:absolute;top:887px;left:213px;white-space:nowrap\" class=\"ft203\">1\/2<\/p>\n<p style=\"position:absolute;top:896px;left:240px;white-space:nowrap\" class=\"ft200\">L=n<\/p>\n<p style=\"position:absolute;top:889px;left:272px;white-space:nowrap\" class=\"ft200\">&pi;&nbsp;<\/p>\n<p style=\"position:absolute;top:898px;left:322px;white-space:nowrap\" class=\"ft200\">Where n=1,2,3 ..<\/p>\n<p style=\"position:absolute;top:897px;left:495px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:898px;left:658px;white-space:nowrap\" class=\"ft200\">(4)<\/p>\n<p style=\"position:absolute;top:897px;left:690px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:916px;left:190px;white-space:nowrap\" class=\"ft200\">&#295;<\/p>\n<p style=\"position:absolute;top:914px;left:201px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:922px;left:211px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:923px;left:284px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:923px;left:561px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:938px;left:108px;white-space:nowrap\" class=\"ft200\">From&nbsp;&nbsp;(4),&nbsp;the energy eigen&nbsp;values&nbsp;may be&nbsp;written&nbsp;as&nbsp;<\/p>\n<p style=\"position:absolute;top:981px;left:108px;white-space:nowrap\" class=\"ft200\">E<\/p>\n<p style=\"position:absolute;top:984px;left:118px;white-space:nowrap\" class=\"ft204\">n<\/p>\n<p style=\"position:absolute;top:981px;left:124px;white-space:nowrap\" class=\"ft200\">&nbsp;&nbsp;=<\/p>\n<p style=\"position:absolute;top:980px;left:154px;white-space:nowrap\" class=\"ft200\">&nbsp;&nbsp;n<\/p>\n<p style=\"position:absolute;top:969px;left:173px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:972px;left:182px;white-space:nowrap\" class=\"ft200\">&pi;<\/p>\n<p style=\"position:absolute;top:969px;left:192px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:978px;left:202px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:971px;left:212px;white-space:nowrap\" class=\"ft200\">&#295;<\/p>\n<p style=\"position:absolute;top:969px;left:223px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:977px;left:233px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:981px;left:258px;white-space:nowrap\" class=\"ft200\">Where n = 1,2,3,&hellip; &hellip;<\/p>\n<p style=\"position:absolute;top:980px;left:463px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:981px;left:668px;white-space:nowrap\" class=\"ft200\">(5)<\/p>\n<p style=\"position:absolute;top:980px;left:701px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1001px;left:173px;white-space:nowrap\" class=\"ft202\">2mL<\/p>\n<p style=\"position:absolute;top:993px;left:204px;white-space:nowrap\" class=\"ft203\">2<\/p>\n<p style=\"position:absolute;top:1000px;left:213px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1008px;left:108px;white-space:nowrap\" class=\"ft200\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1020px;left:108px;white-space:nowrap\" class=\"ft2012\">From&nbsp;this&nbsp;equation, we&nbsp;infer&nbsp;that the&nbsp;energy&nbsp;of the&nbsp;particle&nbsp;is&nbsp;discrete&nbsp;as&nbsp;n&nbsp;can&nbsp;have&nbsp;integer&nbsp;<br \/>values.&nbsp;In other&nbsp;words, the&nbsp;energy&nbsp;is&nbsp;quantized.&nbsp;We&nbsp;also&nbsp;note&nbsp;that n cannot be&nbsp;zero&nbsp;because&nbsp;in&nbsp;<br \/>that case, the&nbsp;wave&nbsp;function as&nbsp;well&nbsp;as&nbsp;the&nbsp;probability&nbsp;of finding&nbsp;the particle becomes&nbsp;zero&nbsp;for&nbsp;<\/p>\n<\/div>\n<div id=\"page21-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224021.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft210\">20&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft217\">&nbsp;<br \/>all&nbsp;values&nbsp;of&nbsp;x.&nbsp;Hence,&nbsp;n&nbsp;= 0&nbsp;is&nbsp;forbidden.&nbsp;The lowest&nbsp;energy the&nbsp;particle can&nbsp;possess&nbsp;is&nbsp;<br \/>corresponding&nbsp;to&nbsp;n =&nbsp;1&nbsp;and is&nbsp;equal&nbsp;to&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:128px;left:194px;white-space:nowrap\" class=\"ft211\">E<\/p>\n<p style=\"position:absolute;top:132px;left:204px;white-space:nowrap\" class=\"ft212\">1<\/p>\n<p style=\"position:absolute;top:128px;left:210px;white-space:nowrap\" class=\"ft211\">&nbsp;=&nbsp;<\/p>\n<p style=\"position:absolute;top:121px;left:228px;white-space:nowrap\" class=\"ft211\">&pi;<\/p>\n<p style=\"position:absolute;top:119px;left:237px;white-space:nowrap\" class=\"ft213\">2<\/p>\n<p style=\"position:absolute;top:121px;left:244px;white-space:nowrap\" class=\"ft211\">&#295;<\/p>\n<p style=\"position:absolute;top:119px;left:253px;white-space:nowrap\" class=\"ft213\">2<\/p>\n<p style=\"position:absolute;top:128px;left:260px;white-space:nowrap\" class=\"ft211\">&nbsp;2mL<\/p>\n<p style=\"position:absolute;top:119px;left:297px;white-space:nowrap\" class=\"ft213\">2<\/p>\n<p style=\"position:absolute;top:127px;left:304px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:153px;left:108px;white-space:nowrap\" class=\"ft219\">&nbsp;<br \/>This&nbsp;is&nbsp;called &lsquo;ground state&nbsp;energy&rsquo; or&nbsp;&lsquo;zero&nbsp;point energy&rsquo;.&nbsp;The&nbsp;higher&nbsp;excited&nbsp;states&nbsp;will have&nbsp;<br \/>energies&nbsp;like&nbsp;4E<\/p>\n<p style=\"position:absolute;top:206px;left:221px;white-space:nowrap\" class=\"ft212\">1<\/p>\n<p style=\"position:absolute;top:200px;left:227px;white-space:nowrap\" class=\"ft210\">,&nbsp;9E<\/p>\n<p style=\"position:absolute;top:206px;left:253px;white-space:nowrap\" class=\"ft212\">1<\/p>\n<p style=\"position:absolute;top:200px;left:259px;white-space:nowrap\" class=\"ft210\">,&nbsp;16E<\/p>\n<p style=\"position:absolute;top:206px;left:295px;white-space:nowrap\" class=\"ft212\">1<\/p>\n<p style=\"position:absolute;top:200px;left:301px;white-space:nowrap\" class=\"ft210\">, etc.&nbsp;This&nbsp;indicates&nbsp;that the&nbsp;energy&nbsp;levels&nbsp;are&nbsp;not equally&nbsp;spaced.&nbsp;<\/p>\n<p style=\"position:absolute;top:216px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:235px;left:108px;white-space:nowrap\" class=\"ft211\">The&nbsp;wave&nbsp;functions&nbsp;or&nbsp;the&nbsp;Eigen functions&nbsp;are&nbsp;<\/p>\n<p style=\"position:absolute;top:234px;left:431px;white-space:nowrap\" class=\"ft210\">given&nbsp;by&nbsp;<\/p>\n<p style=\"position:absolute;top:257px;left:108px;white-space:nowrap\" class=\"ft2110\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:305px;left:162px;white-space:nowrap\" class=\"ft210\">&psi;<\/p>\n<p style=\"position:absolute;top:315px;left:174px;white-space:nowrap\" class=\"ft212\">n<\/p>\n<p style=\"position:absolute;top:311px;left:182px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:311px;left:204px;white-space:nowrap\" class=\"ft210\">= A. Sin<\/p>\n<p style=\"position:absolute;top:310px;left:290px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:308px;left:312px;white-space:nowrap\" class=\"ft210\">2mE<\/p>\n<p style=\"position:absolute;top:312px;left:343px;white-space:nowrap\" class=\"ft212\">n<\/p>\n<p style=\"position:absolute;top:299px;left:350px;white-space:nowrap\" class=\"ft214\">1\/2<\/p>\n<p style=\"position:absolute;top:307px;left:377px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;x&nbsp;<\/p>\n<p style=\"position:absolute;top:310px;left:565px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:335px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:335px;left:198px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:335px;left:227px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:328px;left:324px;white-space:nowrap\" class=\"ft210\">&#295;<\/p>\n<p style=\"position:absolute;top:326px;left:335px;white-space:nowrap\" class=\"ft214\">2<\/p>\n<p style=\"position:absolute;top:334px;left:344px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:335px;left:380px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:335px;left:565px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:363px;left:119px;white-space:nowrap\" class=\"ft210\">or&nbsp; &nbsp;&psi;<\/p>\n<p style=\"position:absolute;top:374px;left:179px;white-space:nowrap\" class=\"ft212\">n<\/p>\n<p style=\"position:absolute;top:369px;left:186px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:370px;left:216px;white-space:nowrap\" class=\"ft210\">=<\/p>\n<p style=\"position:absolute;top:369px;left:227px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;A.&nbsp;Sin&nbsp;<\/p>\n<p style=\"position:absolute;top:370px;left:333px;white-space:nowrap\" class=\"ft210\">n<\/p>\n<p style=\"position:absolute;top:363px;left:344px;white-space:nowrap\" class=\"ft210\">&pi;&nbsp;x&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:370px;left:656px;white-space:nowrap\" class=\"ft210\">(6)<\/p>\n<p style=\"position:absolute;top:369px;left:689px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:391px;left:108px;white-space:nowrap\" class=\"ft211\">&nbsp;<\/p>\n<p style=\"position:absolute;top:391px;left:198px;white-space:nowrap\" class=\"ft211\">&nbsp;<\/p>\n<p style=\"position:absolute;top:391px;left:227px;white-space:nowrap\" class=\"ft211\">&nbsp;<\/p>\n<p style=\"position:absolute;top:391px;left:335px;white-space:nowrap\" class=\"ft210\">L<\/p>\n<p style=\"position:absolute;top:390px;left:345px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:391px;left:380px;white-space:nowrap\" class=\"ft211\">&nbsp;<\/p>\n<p style=\"position:absolute;top:391px;left:565px;white-space:nowrap\" class=\"ft211\">&nbsp;<\/p>\n<p style=\"position:absolute;top:434px;left:108px;white-space:nowrap\" class=\"ft210\">Applying&nbsp;the&nbsp;normalization condition,&nbsp;<\/p>\n<p style=\"position:absolute;top:441px;left:565px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:479px;left:119px;white-space:nowrap\" class=\"ft210\">i.e. &nbsp;<\/p>\n<p style=\"position:absolute;top:472px;left:183px;white-space:nowrap\" class=\"ft210\">&int;&nbsp;A<\/p>\n<p style=\"position:absolute;top:469px;left:210px;white-space:nowrap\" class=\"ft214\">2<\/p>\n<p style=\"position:absolute;top:478px;left:219px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;Sin<\/p>\n<p style=\"position:absolute;top:469px;left:260px;white-space:nowrap\" class=\"ft214\">2<\/p>\n<p style=\"position:absolute;top:479px;left:270px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;n<\/p>\n<p style=\"position:absolute;top:472px;left:302px;white-space:nowrap\" class=\"ft210\">&pi;x&nbsp;. dx&nbsp;&nbsp;= &nbsp;1&nbsp;<\/p>\n<p style=\"position:absolute;top:481px;left:656px;white-space:nowrap\" class=\"ft210\">(7)<\/p>\n<p style=\"position:absolute;top:480px;left:689px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:500px;left:108px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:500px;left:198px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:497px;left:281px;white-space:nowrap\" class=\"ft210\">L<\/p>\n<p style=\"position:absolute;top:496px;left:291px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:500px;left:312px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:500px;left:380px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:500px;left:565px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:507px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:520px;left:108px;white-space:nowrap\" class=\"ft210\">Since&nbsp;the&nbsp;wave&nbsp;function is&nbsp;non-vanishing&nbsp;only&nbsp;for&nbsp;<\/p>\n<p style=\"position:absolute;top:542px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:570px;left:127px;white-space:nowrap\" class=\"ft210\">0&nbsp;&lt;&nbsp;x&nbsp;&lt;&nbsp;L, it can be shown that<\/p>\n<p style=\"position:absolute;top:569px;left:462px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:570px;left:609px;white-space:nowrap\" class=\"ft211\">&nbsp;<\/p>\n<p style=\"position:absolute;top:631px;left:135px;white-space:nowrap\" class=\"ft210\">&int;&nbsp;Sin<\/p>\n<p style=\"position:absolute;top:629px;left:183px;white-space:nowrap\" class=\"ft214\">2<\/p>\n<p style=\"position:absolute;top:638px;left:193px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;n<\/p>\n<p style=\"position:absolute;top:631px;left:225px;white-space:nowrap\" class=\"ft210\">&pi;x&nbsp;&nbsp;dx&nbsp;<\/p>\n<p style=\"position:absolute;top:640px;left:311px;white-space:nowrap\" class=\"ft210\">= (L&nbsp;)<\/p>\n<p style=\"position:absolute;top:639px;left:376px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:640px;left:659px;white-space:nowrap\" class=\"ft210\">(8)<\/p>\n<p style=\"position:absolute;top:639px;left:692px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:657px;left:224px;white-space:nowrap\" class=\"ft210\">L<\/p>\n<p style=\"position:absolute;top:656px;left:235px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:660px;left:246px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:657px;left:343px;white-space:nowrap\" class=\"ft210\">2<\/p>\n<p style=\"position:absolute;top:656px;left:354px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:660px;left:609px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:688px;left:108px;white-space:nowrap\" class=\"ft210\">Substituting&nbsp;in equation (8), we&nbsp;have&nbsp;<\/p>\n<p style=\"position:absolute;top:695px;left:609px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:760px;left:119px;white-space:nowrap\" class=\"ft210\">A<\/p>\n<p style=\"position:absolute;top:751px;left:129px;white-space:nowrap\" class=\"ft214\">2<\/p>\n<p style=\"position:absolute;top:760px;left:137px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;(L&nbsp;) &nbsp;=&nbsp;1<\/p>\n<p style=\"position:absolute;top:759px;left:243px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;or&nbsp;<\/p>\n<p style=\"position:absolute;top:760px;left:302px;white-space:nowrap\" class=\"ft210\">A =&nbsp;(&nbsp;2&nbsp;)<\/p>\n<p style=\"position:absolute;top:751px;left:398px;white-space:nowrap\" class=\"ft214\">1\/2<\/p>\n<p style=\"position:absolute;top:759px;left:426px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:762px;left:659px;white-space:nowrap\" class=\"ft210\">(9)<\/p>\n<p style=\"position:absolute;top:762px;left:692px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:779px;left:160px;white-space:nowrap\" class=\"ft210\">2<\/p>\n<p style=\"position:absolute;top:778px;left:171px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:782px;left:246px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:779px;left:376px;white-space:nowrap\" class=\"ft210\">L<\/p>\n<p style=\"position:absolute;top:778px;left:387px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:782px;left:609px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:786px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:795px;left:108px;white-space:nowrap\" class=\"ft210\">The&nbsp;eigen function or&nbsp;wave&nbsp;functions&nbsp;in equation&nbsp;(9) becomes&nbsp;<\/p>\n<p style=\"position:absolute;top:816px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:845px;left:162px;white-space:nowrap\" class=\"ft210\">&psi;<\/p>\n<p style=\"position:absolute;top:856px;left:174px;white-space:nowrap\" class=\"ft212\">n<\/p>\n<p style=\"position:absolute;top:851px;left:182px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;= (&nbsp;2&nbsp;)<\/p>\n<p style=\"position:absolute;top:839px;left:268px;white-space:nowrap\" class=\"ft214\">&frac12;<\/p>\n<p style=\"position:absolute;top:848px;left:277px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;sin&nbsp;&nbsp;(2mE<\/p>\n<p style=\"position:absolute;top:852px;left:384px;white-space:nowrap\" class=\"ft212\">n<\/p>\n<p style=\"position:absolute;top:849px;left:391px;white-space:nowrap\" class=\"ft210\">)<\/p>\n<p style=\"position:absolute;top:839px;left:402px;white-space:nowrap\" class=\"ft214\">&frac12;<\/p>\n<p style=\"position:absolute;top:849px;left:411px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;x<\/p>\n<p style=\"position:absolute;top:848px;left:444px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:851px;left:536px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:885px;left:162px;white-space:nowrap\" class=\"ft210\">&psi;<\/p>\n<p style=\"position:absolute;top:896px;left:174px;white-space:nowrap\" class=\"ft212\">n<\/p>\n<p style=\"position:absolute;top:891px;left:182px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;= (&nbsp;2&nbsp;)<\/p>\n<p style=\"position:absolute;top:880px;left:268px;white-space:nowrap\" class=\"ft214\">&frac12;<\/p>\n<p style=\"position:absolute;top:888px;left:277px;white-space:nowrap\" class=\"ft210\">&nbsp;&nbsp;sin&nbsp;&nbsp;n<\/p>\n<p style=\"position:absolute;top:884px;left:351px;white-space:nowrap\" class=\"ft210\">&pi;x&nbsp;<\/p>\n<p style=\"position:absolute;top:911px;left:162px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:911px;left:187px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:908px;left:237px;white-space:nowrap\" class=\"ft210\">L<\/p>\n<p style=\"position:absolute;top:907px;left:248px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:911px;left:281px;white-space:nowrap\" class=\"ft212\">&nbsp;<\/p>\n<p style=\"position:absolute;top:908px;left:357px;white-space:nowrap\" class=\"ft210\">L<\/p>\n<p style=\"position:absolute;top:907px;left:368px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:920px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:942px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:964px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:986px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1008px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1030px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1052px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1074px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1096px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1118px;left:108px;white-space:nowrap\" class=\"ft210\">&nbsp;<\/p>\n<\/div>\n<div id=\"page22-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224022.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft220\">21&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft227\">&nbsp;<br \/>&nbsp; &nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:68px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:90px;left:108px;white-space:nowrap\" class=\"ft228\">Fig.&nbsp;(2)&nbsp;shows&nbsp;the&nbsp;variation of the&nbsp;wave&nbsp;function inside&nbsp;the&nbsp;box&nbsp;for&nbsp;different values&nbsp;of n and&nbsp;<br \/>Fig.(3)&nbsp;shows&nbsp;the&nbsp;probability&nbsp;densities&nbsp;of finding&nbsp;the&nbsp;particle&nbsp;at &nbsp;different&nbsp; places&nbsp;<\/p>\n<p style=\"position:absolute;top:120px;left:702px;white-space:nowrap\" class=\"ft221\">inside&nbsp; the&nbsp; box&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:149px;left:108px;white-space:nowrap\" class=\"ft221\">for &nbsp;different&nbsp;<\/p>\n<p style=\"position:absolute;top:150px;left:205px;white-space:nowrap\" class=\"ft222\">values&nbsp; of&nbsp; n. &nbsp;&nbsp;Thus,&nbsp; wave &nbsp;mechanics&nbsp;suggests&nbsp;&nbsp;that&nbsp;&nbsp;the&nbsp;<\/p>\n<p style=\"position:absolute;top:149px;left:613px;white-space:nowrap\" class=\"ft221\">probability&nbsp; of&nbsp; finding&nbsp; any&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:178px;left:108px;white-space:nowrap\" class=\"ft221\">particle&nbsp; at&nbsp; the&nbsp; lowest&nbsp;<\/p>\n<p style=\"position:absolute;top:178px;left:278px;white-space:nowrap\" class=\"ft220\">energy&nbsp;level&nbsp;is&nbsp;maximum&nbsp;at the&nbsp;centre&nbsp;of the&nbsp;box&nbsp;which is&nbsp;in agreement&nbsp;<\/p>\n<p style=\"position:absolute;top:207px;left:108px;white-space:nowrap\" class=\"ft228\">with the&nbsp;classical&nbsp;picture.&nbsp;However, the&nbsp;probability&nbsp;of finding&nbsp;the&nbsp;particle&nbsp;in higher&nbsp;energy&nbsp;states&nbsp;<br \/>is&nbsp;predicted&nbsp;differently&nbsp;by&nbsp;the&nbsp;two&nbsp;formulations.&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:289px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:311px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:333px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:355px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:377px;left:147px;white-space:nowrap\" class=\"ft220\">S.NO&nbsp;<\/p>\n<p style=\"position:absolute;top:377px;left:317px;white-space:nowrap\" class=\"ft220\">RGPV QUESTION&nbsp;<\/p>\n<p style=\"position:absolute;top:377px;left:582px;white-space:nowrap\" class=\"ft220\">YEAR&nbsp;<\/p>\n<p style=\"position:absolute;top:377px;left:712px;white-space:nowrap\" class=\"ft220\">MARKS&nbsp;<\/p>\n<p style=\"position:absolute;top:400px;left:116px;white-space:nowrap\" class=\"ft220\">Q.1&nbsp;<\/p>\n<p style=\"position:absolute;top:400px;left:231px;white-space:nowrap\" class=\"ft220\">Obtain an expression of energy&nbsp;levels&nbsp;<\/p>\n<p style=\"position:absolute;top:422px;left:231px;white-space:nowrap\" class=\"ft220\">for&nbsp;particle&nbsp;trapped in&nbsp;one&nbsp;dimensional&nbsp;<\/p>\n<p style=\"position:absolute;top:444px;left:231px;white-space:nowrap\" class=\"ft220\">square&nbsp;with infinitely&nbsp;deep potential&nbsp;<\/p>\n<p style=\"position:absolute;top:466px;left:231px;white-space:nowrap\" class=\"ft220\">well.&nbsp;<\/p>\n<p style=\"position:absolute;top:400px;left:569px;white-space:nowrap\" class=\"ft220\">Dec2013&nbsp;<\/p>\n<p style=\"position:absolute;top:400px;left:734px;white-space:nowrap\" class=\"ft220\">7&nbsp;<\/p>\n<p style=\"position:absolute;top:489px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:511px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:533px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:555px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:577px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:600px;left:379px;white-space:nowrap\" class=\"ft223\"><b>UNIT&nbsp;1\/LECTURE&nbsp;8&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:626px;left:108px;white-space:nowrap\" class=\"ft224\"><b>&nbsp;&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:648px;left:108px;white-space:nowrap\" class=\"ft224\"><b>&nbsp; &nbsp;<\/b>Q1.&nbsp;X-rays&nbsp;&nbsp;of &nbsp;wavelength &nbsp;1.54&nbsp; A<\/p>\n<p style=\"position:absolute;top:646px;left:372px;white-space:nowrap\" class=\"ft225\">o<\/p>\n<p style=\"position:absolute;top:648px;left:378px;white-space:nowrap\" class=\"ft220\">&nbsp;&nbsp;are&nbsp; Compton &nbsp;scattered at &nbsp;an angle&nbsp;of 60<\/p>\n<p style=\"position:absolute;top:646px;left:695px;white-space:nowrap\" class=\"ft225\">o<\/p>\n<p style=\"position:absolute;top:648px;left:701px;white-space:nowrap\" class=\"ft220\">.&nbsp;&nbsp;Calculate&nbsp;the&nbsp;<\/p>\n<p style=\"position:absolute;top:670px;left:108px;white-space:nowrap\" class=\"ft220\">change&nbsp;in the&nbsp;wavelength.&nbsp;<\/p>\n<p style=\"position:absolute;top:684px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:702px;left:108px;white-space:nowrap\" class=\"ft224\"><b>Solution:<\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:714px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:731px;left:104px;white-space:nowrap\" class=\"ft220\">Change&nbsp;in wavelength&nbsp;=&nbsp;&#8710;&lambda;&nbsp;=&nbsp;h&nbsp;(1-cos&nbsp;&theta;) \/m<\/p>\n<p style=\"position:absolute;top:742px;left:500px;white-space:nowrap\" class=\"ft225\">o<\/p>\n<p style=\"position:absolute;top:738px;left:507px;white-space:nowrap\" class=\"ft220\">c<\/p>\n<p style=\"position:absolute;top:737px;left:518px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:753px;left:108px;white-space:nowrap\" class=\"ft229\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:794px;left:104px;white-space:nowrap\" class=\"ft220\">h&nbsp;(1-cos 60<\/p>\n<p style=\"position:absolute;top:785px;left:223px;white-space:nowrap\" class=\"ft226\">0<\/p>\n<p style=\"position:absolute;top:794px;left:233px;white-space:nowrap\" class=\"ft220\">)\/ m<\/p>\n<p style=\"position:absolute;top:798px;left:276px;white-space:nowrap\" class=\"ft225\">o<\/p>\n<p style=\"position:absolute;top:794px;left:283px;white-space:nowrap\" class=\"ft220\">c<\/p>\n<p style=\"position:absolute;top:793px;left:294px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:811px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:830px;left:148px;white-space:nowrap\" class=\"ft220\">=<\/p>\n<p style=\"position:absolute;top:829px;left:158px;white-space:nowrap\" class=\"ft220\">&nbsp;&nbsp;= 1.2&nbsp;x 10<\/p>\n<p style=\"position:absolute;top:821px;left:277px;white-space:nowrap\" class=\"ft226\">-12<\/p>\n<p style=\"position:absolute;top:830px;left:305px;white-space:nowrap\" class=\"ft220\">m (Ans).&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:848px;left:108px;white-space:nowrap\" class=\"ft229\">&nbsp;<br \/>&nbsp;<br \/><a href=\"https:\/\/www.rgpvonline.com\/\" target=\"_blank\" rel=\"noopener\">Q2.&nbsp;<\/a><\/p>\n<p style=\"position:absolute;top:876px;left:139px;white-space:nowrap\" class=\"ft220\"><a href=\"https:\/\/www.rgpvonline.com\/\" target=\"_blank\" rel=\"noopener\">In&nbsp;&nbsp;a&nbsp;&nbsp;Compton&nbsp;&nbsp;scattering&nbsp; experiment,&nbsp; incident&nbsp; photons&nbsp;of&nbsp;energy&nbsp;10&nbsp;KeV&nbsp;are&nbsp;scattered&nbsp;at&nbsp;<\/a><\/p>\n<p style=\"position:absolute;top:898px;left:108px;white-space:nowrap\" class=\"ft220\"><a href=\"https:\/\/www.rgpvonline.com\/\" target=\"_blank\" rel=\"noopener\">45<\/a><\/p>\n<p style=\"position:absolute;top:896px;left:126px;white-space:nowrap\" class=\"ft225\"><a href=\"https:\/\/www.rgpvonline.com\/\" target=\"_blank\" rel=\"noopener\">o<\/a><\/p>\n<p style=\"position:absolute;top:898px;left:132px;white-space:nowrap\" class=\"ft220\"><a href=\"https:\/\/www.rgpvonline.com\/\" target=\"_blank\" rel=\"noopener\">&nbsp;to&nbsp;the&nbsp;incident beam.&nbsp;Calculate&nbsp;the&nbsp;energy&nbsp;of&nbsp;the&nbsp;scattered photon.&nbsp;<\/a><\/p>\n<p style=\"position:absolute;top:906px;left:108px;white-space:nowrap\" class=\"ft220\"><a href=\"https:\/\/www.rgpvonline.com\/\" target=\"_blank\" rel=\"noopener\">&nbsp;<\/a><\/p>\n<p style=\"position:absolute;top:924px;left:108px;white-space:nowrap\" class=\"ft224\"><b>Solution:<\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:935px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:952px;left:108px;white-space:nowrap\" class=\"ft220\">Change&nbsp;in&nbsp;wavelength&nbsp;=&nbsp;&#8710;&lambda;&nbsp;=&nbsp;h&nbsp;(1-cos&nbsp;&theta;) m<\/p>\n<p style=\"position:absolute;top:963px;left:493px;white-space:nowrap\" class=\"ft225\">o<\/p>\n<p style=\"position:absolute;top:959px;left:500px;white-space:nowrap\" class=\"ft220\">c<\/p>\n<p style=\"position:absolute;top:958px;left:511px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:966px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:988px;left:346px;white-space:nowrap\" class=\"ft220\">= 7.1 x 10<\/p>\n<p style=\"position:absolute;top:979px;left:454px;white-space:nowrap\" class=\"ft226\">-13<\/p>\n<p style=\"position:absolute;top:988px;left:482px;white-space:nowrap\" class=\"ft220\">m.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:1010px;left:346px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1026px;left:108px;white-space:nowrap\" class=\"ft220\">Wavelength&nbsp;of incident photon&nbsp;=&nbsp;&lambda;&nbsp;= hc\/eE&nbsp;<\/p>\n<p style=\"position:absolute;top:1049px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1071px;left:475px;white-space:nowrap\" class=\"ft220\">= 1.243 x 10<\/p>\n<p style=\"position:absolute;top:1062px;left:605px;white-space:nowrap\" class=\"ft226\">10<\/p>\n<p style=\"position:absolute;top:1071px;left:624px;white-space:nowrap\" class=\"ft220\">m.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:1088px;left:108px;white-space:nowrap\" class=\"ft220\">Wavelength&nbsp;of scattered photon&nbsp;=&lambda;&rsquo;=&nbsp;&lambda;&nbsp;+&nbsp;&#8710;&lambda;&nbsp;<\/p>\n<p style=\"position:absolute;top:1111px;left:108px;white-space:nowrap\" class=\"ft220\">&nbsp;<\/p>\n<\/div>\n<div id=\"page23-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224023.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft230\">22&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:54px;left:443px;white-space:nowrap\" class=\"ft230\">= 1.25 x 10<\/p>\n<p style=\"position:absolute;top:45px;left:562px;white-space:nowrap\" class=\"ft231\">10<\/p>\n<p style=\"position:absolute;top:54px;left:581px;white-space:nowrap\" class=\"ft230\">m.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:71px;left:108px;white-space:nowrap\" class=\"ft230\">Energy&nbsp;of scattered photon&nbsp;= hc\/&lambda;&rsquo;&nbsp;<\/p>\n<p style=\"position:absolute;top:92px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:116px;left:400px;white-space:nowrap\" class=\"ft230\">=<\/p>\n<p style=\"position:absolute;top:114px;left:410px;white-space:nowrap\" class=\"ft230\">&nbsp;&nbsp;1.59 x 10<\/p>\n<p style=\"position:absolute;top:107px;left:519px;white-space:nowrap\" class=\"ft231\">-15<\/p>\n<p style=\"position:absolute;top:116px;left:547px;white-space:nowrap\" class=\"ft230\">&nbsp;J&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:123px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:139px;left:400px;white-space:nowrap\" class=\"ft230\">=<\/p>\n<p style=\"position:absolute;top:137px;left:410px;white-space:nowrap\" class=\"ft230\">&nbsp;&nbsp;9.93 keV (Ans).&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:156px;left:108px;white-space:nowrap\" class=\"ft236\">&nbsp;<br \/>&nbsp;<br \/>Q3.<\/p>\n<p style=\"position:absolute;top:184px;left:134px;white-space:nowrap\" class=\"ft230\">Gamma&nbsp;Rays&nbsp;of&nbsp;energy 0.5&nbsp;MeV&nbsp;are scattered&nbsp;by electrons.&nbsp;What&nbsp;is&nbsp;the energy&nbsp;of&nbsp;scattered&nbsp;<\/p>\n<p style=\"position:absolute;top:210px;left:108px;white-space:nowrap\" class=\"ft230\">gamma rays&nbsp; at&nbsp;a scattering&nbsp;angle&nbsp;of&nbsp;30<\/p>\n<p style=\"position:absolute;top:208px;left:392px;white-space:nowrap\" class=\"ft232\">o<\/p>\n<p style=\"position:absolute;top:210px;left:398px;white-space:nowrap\" class=\"ft230\">?&nbsp;What is&nbsp;the&nbsp;kinetic&nbsp;energy&nbsp;of scattered electron?&nbsp;<\/p>\n<p style=\"position:absolute;top:225px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:243px;left:111px;white-space:nowrap\" class=\"ft233\"><b>Solution:<\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:257px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:278px;left:111px;white-space:nowrap\" class=\"ft230\">Wavelength&nbsp;of incident gamma&nbsp;rays&nbsp;=&nbsp;&lambda;&nbsp;= hc\/E&nbsp;<\/p>\n<p style=\"position:absolute;top:299px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:323px;left:316px;white-space:nowrap\" class=\"ft230\">=<\/p>\n<p style=\"position:absolute;top:321px;left:327px;white-space:nowrap\" class=\"ft230\">&nbsp;&nbsp;6.62&#215;10<\/p>\n<p style=\"position:absolute;top:314px;left:413px;white-space:nowrap\" class=\"ft231\">-34<\/p>\n<p style=\"position:absolute;top:323px;left:441px;white-space:nowrap\" class=\"ft230\">x3x10<\/p>\n<p style=\"position:absolute;top:314px;left:495px;white-space:nowrap\" class=\"ft231\">8<\/p>\n<p style=\"position:absolute;top:323px;left:505px;white-space:nowrap\" class=\"ft230\">\/1.6&#215;10<\/p>\n<p style=\"position:absolute;top:314px;left:581px;white-space:nowrap\" class=\"ft231\">-19<\/p>\n<p style=\"position:absolute;top:323px;left:609px;white-space:nowrap\" class=\"ft230\">x0.5&#215;10<\/p>\n<p style=\"position:absolute;top:314px;left:684px;white-space:nowrap\" class=\"ft231\">6<\/p>\n<p style=\"position:absolute;top:323px;left:694px;white-space:nowrap\" class=\"ft230\">&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:333px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:353px;left:302px;white-space:nowrap\" class=\"ft230\">=<\/p>\n<p style=\"position:absolute;top:352px;left:313px;white-space:nowrap\" class=\"ft230\">&nbsp;&nbsp;2.486 x 10<\/p>\n<p style=\"position:absolute;top:344px;left:432px;white-space:nowrap\" class=\"ft231\">-12<\/p>\n<p style=\"position:absolute;top:353px;left:460px;white-space:nowrap\" class=\"ft230\">m.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:361px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:373px;left:108px;white-space:nowrap\" class=\"ft230\">Change&nbsp;in wavelength&nbsp;&nbsp;=&nbsp;<\/p>\n<p style=\"position:absolute;top:374px;left:362px;white-space:nowrap\" class=\"ft230\">&#8710;&lambda;&nbsp;=&nbsp;h&nbsp;&nbsp;(1-cos&nbsp;&theta;)&nbsp;<\/p>\n<p style=\"position:absolute;top:401px;left:108px;white-space:nowrap\" class=\"ft234\">&nbsp;<\/p>\n<p style=\"position:absolute;top:401px;left:435px;white-space:nowrap\" class=\"ft230\">m<\/p>\n<p style=\"position:absolute;top:404px;left:446px;white-space:nowrap\" class=\"ft232\">o<\/p>\n<p style=\"position:absolute;top:401px;left:453px;white-space:nowrap\" class=\"ft230\">c<\/p>\n<p style=\"position:absolute;top:400px;left:464px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:431px;left:351px;white-space:nowrap\" class=\"ft230\">=<\/p>\n<p style=\"position:absolute;top:430px;left:362px;white-space:nowrap\" class=\"ft230\">&nbsp;3.24 x 10<\/p>\n<p style=\"position:absolute;top:419px;left:463px;white-space:nowrap\" class=\"ft231\">-13<\/p>\n<p style=\"position:absolute;top:428px;left:492px;white-space:nowrap\" class=\"ft230\">m.<\/p>\n<p style=\"position:absolute;top:427px;left:513px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:443px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:458px;left:111px;white-space:nowrap\" class=\"ft230\">Wavelength&nbsp;of scattered photon&nbsp;=&lambda;&rsquo;=&nbsp;&lambda;&nbsp;+&nbsp;&#8710;&lambda;&nbsp;<\/p>\n<p style=\"position:absolute;top:481px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:505px;left:446px;white-space:nowrap\" class=\"ft234\">2.81 x 10<\/p>\n<p style=\"position:absolute;top:497px;left:539px;white-space:nowrap\" class=\"ft231\">-12<\/p>\n<p style=\"position:absolute;top:505px;left:566px;white-space:nowrap\" class=\"ft234\">m.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:524px;left:111px;white-space:nowrap\" class=\"ft234\">Kinetic&nbsp;energy&nbsp;of the scattered&nbsp;electron&nbsp;= hc\/&lambda;&rsquo;<\/p>\n<p style=\"position:absolute;top:529px;left:469px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:543px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:561px;left:402px;white-space:nowrap\" class=\"ft230\">= 0.442 MeV (Ans).<\/p>\n<p style=\"position:absolute;top:560px;left:596px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:577px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:602px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:625px;left:108px;white-space:nowrap\" class=\"ft234\">Q4. &nbsp;<\/p>\n<p style=\"position:absolute;top:619px;left:142px;white-space:nowrap\" class=\"ft234\">X-rays&nbsp; &nbsp;of&nbsp;&nbsp;&nbsp;wavelength&nbsp; &nbsp;1.5 &nbsp;&nbsp;A<\/p>\n<p style=\"position:absolute;top:617px;left:358px;white-space:nowrap\" class=\"ft232\">o<\/p>\n<p style=\"position:absolute;top:619px;left:364px;white-space:nowrap\" class=\"ft234\">&nbsp; &nbsp;are&nbsp; &nbsp;Compton&nbsp;<\/p>\n<p style=\"position:absolute;top:618px;left:478px;white-space:nowrap\" class=\"ft230\">scattered.&nbsp;At what angle&nbsp;will&nbsp;be&nbsp;scattered x-<\/p>\n<p style=\"position:absolute;top:640px;left:108px;white-space:nowrap\" class=\"ft230\">rays&nbsp;have&nbsp;a&nbsp;wavelength of&nbsp;1.506 A&#61548;?&nbsp;<\/p>\n<p style=\"position:absolute;top:650px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:668px;left:111px;white-space:nowrap\" class=\"ft233\"><b>Solution:<\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:680px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:697px;left:111px;white-space:nowrap\" class=\"ft230\">Change&nbsp;in wavelength&nbsp;=&nbsp;&#8710;&lambda;&nbsp;=&nbsp;h&nbsp;(1-cos&nbsp;&theta;) m<\/p>\n<p style=\"position:absolute;top:708px;left:496px;white-space:nowrap\" class=\"ft232\">o<\/p>\n<p style=\"position:absolute;top:704px;left:503px;white-space:nowrap\" class=\"ft230\">c<\/p>\n<p style=\"position:absolute;top:703px;left:514px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:713px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:734px;left:111px;white-space:nowrap\" class=\"ft230\">cos&nbsp;<\/p>\n<p style=\"position:absolute;top:727px;left:154px;white-space:nowrap\" class=\"ft230\">&theta;&nbsp;= (1&nbsp;&ndash;&nbsp;m<\/p>\n<p style=\"position:absolute;top:738px;left:261px;white-space:nowrap\" class=\"ft232\">0<\/p>\n<p style=\"position:absolute;top:734px;left:268px;white-space:nowrap\" class=\"ft230\">c.&nbsp;<\/p>\n<p style=\"position:absolute;top:727px;left:300px;white-space:nowrap\" class=\"ft230\">&#8710;&lambda;\/h) = (1&nbsp;&ndash;&nbsp;0.247) =0.753&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:759px;left:111px;white-space:nowrap\" class=\"ft230\">Angle&nbsp;of&nbsp;scattering,&theta;&nbsp;= 41.2<\/p>\n<p style=\"position:absolute;top:757px;left:343px;white-space:nowrap\" class=\"ft231\">0<\/p>\n<p style=\"position:absolute;top:766px;left:353px;white-space:nowrap\" class=\"ft230\">&nbsp;(Ans).<\/p>\n<p style=\"position:absolute;top:765px;left:428px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:794px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:809px;left:108px;white-space:nowrap\" class=\"ft237\">&nbsp;<br \/>Q5.<\/p>\n<p style=\"position:absolute;top:823px;left:134px;white-space:nowrap\" class=\"ft230\">Calculate&nbsp;the&nbsp;de&nbsp;Broglie&nbsp;wavelength associated with an electron travelling&nbsp;with a&nbsp;velocity&nbsp;of&nbsp;<\/p>\n<p style=\"position:absolute;top:855px;left:108px;white-space:nowrap\" class=\"ft230\">10<\/p>\n<p style=\"position:absolute;top:846px;left:126px;white-space:nowrap\" class=\"ft231\">5<\/p>\n<p style=\"position:absolute;top:855px;left:134px;white-space:nowrap\" class=\"ft230\">&nbsp;ms<\/p>\n<p style=\"position:absolute;top:846px;left:160px;white-space:nowrap\" class=\"ft231\">-1<\/p>\n<p style=\"position:absolute;top:855px;left:173px;white-space:nowrap\" class=\"ft230\">.&nbsp;Assume&nbsp;the&nbsp;mass&nbsp;of the&nbsp;electron&nbsp;to be&nbsp;9.1&nbsp;x&nbsp;10<\/p>\n<p style=\"position:absolute;top:846px;left:523px;white-space:nowrap\" class=\"ft231\">-31<\/p>\n<p style=\"position:absolute;top:855px;left:544px;white-space:nowrap\" class=\"ft230\">kg.&nbsp;and&nbsp;h&nbsp;=&nbsp;6.62&#215;10<\/p>\n<p style=\"position:absolute;top:846px;left:685px;white-space:nowrap\" class=\"ft231\">-34<\/p>\n<p style=\"position:absolute;top:855px;left:706px;white-space:nowrap\" class=\"ft230\">Js.&nbsp;<\/p>\n<p style=\"position:absolute;top:883px;left:108px;white-space:nowrap\" class=\"ft233\"><b>Solution:<\/b>&nbsp;<\/p>\n<p style=\"position:absolute;top:884px;left:432px;white-space:nowrap\" class=\"ft234\">&nbsp;<\/p>\n<p style=\"position:absolute;top:922px;left:129px;white-space:nowrap\" class=\"ft230\">De&nbsp;Broglie&nbsp;wavelength&nbsp;&lambda;&nbsp;=&nbsp;h&nbsp;<\/p>\n<p style=\"position:absolute;top:927px;left:439px;white-space:nowrap\" class=\"ft230\">= &nbsp;6.62 x 10<\/p>\n<p style=\"position:absolute;top:918px;left:562px;white-space:nowrap\" class=\"ft231\">-34<\/p>\n<p style=\"position:absolute;top:927px;left:589px;white-space:nowrap\" class=\"ft230\">____<\/p>\n<p style=\"position:absolute;top:926px;left:630px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:956px;left:411px;white-space:nowrap\" class=\"ft230\">P<\/p>\n<p style=\"position:absolute;top:955px;left:422px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:954px;left:461px;white-space:nowrap\" class=\"ft230\">9.1 x 10<\/p>\n<p style=\"position:absolute;top:945px;left:542px;white-space:nowrap\" class=\"ft231\">-31<\/p>\n<p style=\"position:absolute;top:954px;left:568px;white-space:nowrap\" class=\"ft230\">x10<\/p>\n<p style=\"position:absolute;top:945px;left:598px;white-space:nowrap\" class=\"ft231\">5<\/p>\n<p style=\"position:absolute;top:953px;left:607px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:972px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:989px;left:378px;white-space:nowrap\" class=\"ft230\">&lambda;&nbsp;&nbsp;= 7.27 x 10<\/p>\n<p style=\"position:absolute;top:987px;left:518px;white-space:nowrap\" class=\"ft231\">-9<\/p>\n<p style=\"position:absolute;top:996px;left:537px;white-space:nowrap\" class=\"ft230\">&nbsp;&nbsp;m. (Ans.)&nbsp;<\/p>\n<p style=\"position:absolute;top:989px;left:666px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1010px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1027px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1047px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1069px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1091px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<p style=\"position:absolute;top:1113px;left:108px;white-space:nowrap\" class=\"ft230\">&nbsp;<\/p>\n<\/div>\n<div id=\"page24-div\" style=\"position:relative;width:918px;height:1188px;\" class=\"kk_zoom\">\n<img width=\"918\" height=\"1188\" src=\"https:\/\/content.kopykitab.com\/pdftohtml\/01c536e431c4f99277130978b38461bf224024.png\" alt=\"background image\"><\/p>\n<p style=\"position:absolute;top:3px;left:450px;white-space:nowrap\" class=\"ft240\">23&nbsp;<\/p>\n<p style=\"position:absolute;top:26px;left:108px;white-space:nowrap\" class=\"ft244\">&nbsp;<br \/>&nbsp;<\/p>\n<p style=\"position:absolute;top:67px;left:108px;white-space:nowrap\" class=\"ft240\">&nbsp;<\/p>\n<p style=\"position:absolute;top:89px;left:287px;white-space:nowrap\" class=\"ft241\"><b>UNIT&nbsp;1\/LECTURE&nbsp;9\/ADDITIONAL&nbsp;TOPICS&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:122px;left:108px;white-space:nowrap\" class=\"ft245\">&nbsp;<br \/><b>Photoelectric effect&nbsp;<\/b><\/p>\n<p style=\"position:absolute;top:156px;left:108px;white-space:nowrap\" class=\"ft240\">&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:158px;left:181px;white-space:nowrap\" class=\"ft243\">The photoelectric&nbsp;effect&nbsp;occurs&nbsp;when&nbsp;matter&nbsp;emits&nbsp;electrons&nbsp;upon&nbsp;exposure to&nbsp;<\/p>\n<p style=\"position:absolute;top:178px;left:108px;white-space:nowrap\" class=\"ft243\">electromagnetic&nbsp;radiation,&nbsp;such as&nbsp;photons&nbsp;of&nbsp;light.&nbsp;Here&#8217;s&nbsp;a&nbsp;closer&nbsp;look&nbsp;at&nbsp;what&nbsp;the&nbsp;photoelectric&nbsp;effect&nbsp;<\/p>\n<p style=\"position:absolute;top:198px;left:108px;white-space:nowrap\" class=\"ft243\">is&nbsp;and&nbsp;how&nbsp;it&nbsp;works.&nbsp;<\/p>\n<p style=\"position:absolute;top:197px;left:245px;white-space:nowrap\" class=\"ft240\">&nbsp;<\/p>\n<p style=\"position:absolute;top:218px;left:108px;white-space:nowrap\" class=\"ft240\">The&nbsp;photoelectric&nbsp;effect&nbsp;is&nbsp;studied in&nbsp;part&nbsp;because&nbsp;it can be&nbsp;an introduction to&nbsp;wave-particle&nbsp;<\/p>\n<p style=\"position:absolute;top:240px;left:108px;white-space:nowrap\" class=\"ft240\">duality&nbsp;and quantum&nbsp;mechanics.&nbsp;<\/p>\n<p style=\"position:absolute;top:283px;left:108px;white-space:nowrap\" class=\"ft240\">When a&nbsp;surface&nbsp;is&nbsp;exposed to&nbsp;sufficiently&nbsp;energetic&nbsp;electromagnetic&nbsp;energy, light will&nbsp;be&nbsp;<\/p>\n<p style=\"position:absolute;top:305px;left:108px;white-space:nowrap\" class=\"ft240\">absorbed and electrons&nbsp;will&nbsp;be&nbsp;emitted.&nbsp;The&nbsp;threshold frequency&nbsp;is&nbsp;different for&nbsp;different&nbsp;<\/p>\n<p style=\"position:absolute;top:327px;left:108px;white-space:nowrap\" class=\"ft240\">materials.&nbsp;It&nbsp;is&nbsp;visible&nbsp;light&nbsp;for alkali metals,&nbsp;near-ultraviolet&nbsp;light&nbsp;for other&nbsp;metals,&nbsp;and&nbsp;extreme-<\/p>\n<p style=\"position:absolute;top:349px;left:108px;white-space:nowrap\" class=\"ft240\">ultraviolet radiation for&nbsp;nonmetals.&nbsp;The&nbsp;photoelectric&nbsp;effect&nbsp;occurs&nbsp;with photons&nbsp;having&nbsp;<\/p>\n<p style=\"position:absolute;top:371px;left:108px;white-space:nowrap\" class=\"ft240\">energies&nbsp;from&nbsp;a&nbsp;few electronvolts&nbsp;to&nbsp;over&nbsp;1&nbsp;MeV.&nbsp;At the&nbsp;high&nbsp;photon&nbsp;energies&nbsp;comparable&nbsp;to&nbsp;<\/p>\n<p style=\"position:absolute;top:393px;left:108px;white-space:nowrap\" class=\"ft240\">the&nbsp;electron rest energy&nbsp;of 511&nbsp;keV,&nbsp;Compton scattering&nbsp;may&nbsp;occur&nbsp;pair&nbsp;production&nbsp;may&nbsp;take&nbsp;<\/p>\n<p style=\"position:absolute;top:415px;left:108px;white-space:nowrap\" class=\"ft240\">place&nbsp;at&nbsp;energies&nbsp;over&nbsp;1.022 MeV.&nbsp;<\/p>\n<p style=\"position:absolute;top:458px;left:108px;white-space:nowrap\" class=\"ft240\">Einstein proposed that light consisted of quanta,&nbsp;which we&nbsp;call&nbsp;photons.&nbsp;He&nbsp;suggested that the&nbsp;<\/p>\n<p style=\"position:absolute;top:480px;left:108px;white-space:nowrap\" class=\"ft240\">energy&nbsp;in each quantum&nbsp;of light was&nbsp;equal&nbsp;to&nbsp;the&nbsp;frequency&nbsp;multiplied by&nbsp;a&nbsp;constant (Planck&#8217;s&nbsp;<\/p>\n<p style=\"position:absolute;top:502px;left:108px;white-space:nowrap\" class=\"ft240\">constant)&nbsp;and that a&nbsp;photon with a&nbsp;frequency&nbsp;over&nbsp;a&nbsp;certain&nbsp;threshold would have&nbsp;sufficient&nbsp;<\/p>\n<p style=\"position:absolute;top:524px;left:108px;white-space:nowrap\" class=\"ft240\">energy&nbsp;to&nbsp;eject a&nbsp;single&nbsp;electron, producing&nbsp;the&nbsp;photoelectric&nbsp;effect.&nbsp;It turns&nbsp;out that light does&nbsp;<\/p>\n<p style=\"position:absolute;top:546px;left:108px;white-space:nowrap\" class=\"ft240\">not need to&nbsp;be&nbsp;quantized in order&nbsp;to&nbsp;explain the&nbsp;photoelectric&nbsp;effect, but some&nbsp;textbooks&nbsp;<\/p>\n<p style=\"position:absolute;top:568px;left:108px;white-space:nowrap\" class=\"ft240\">persist in saying&nbsp;that the&nbsp;photoelectric&nbsp;effect demonstrates&nbsp;the&nbsp;particle&nbsp;nature&nbsp;of light.&nbsp;&nbsp;<\/p>\n<p style=\"position:absolute;top:611px;left:108px;white-space:nowrap\" class=\"ft245\">&nbsp;<br \/>&nbsp;<br \/>&nbsp;<br \/>&nbsp;<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>1&nbsp; &nbsp; UNIT&nbsp;&ndash;&nbsp;1&nbsp;&nbsp; &nbsp; QUANTUM&nbsp;PHYSICS&nbsp; Unit-01\/Lecture-01&nbsp; &nbsp; Concept&nbsp;of&nbsp;matter&nbsp;waves&nbsp; Louis&nbsp;de&nbsp;Broglie&nbsp;made&nbsp;the&nbsp;suggestion that particles&nbsp;of matter,&nbsp;like&nbsp;electrons, might possess&nbsp;wave&nbsp;properties&nbsp;and hence&nbsp;exhibit dual&nbsp;nature.&nbsp;His&nbsp;hypothesis&nbsp;was&nbsp;based on the&nbsp;following&nbsp;arguments:&nbsp; &nbsp; The&nbsp;Planck&rsquo;s&nbsp; theory&nbsp; of &nbsp;radiation &nbsp;suggests&nbsp; that &nbsp;energy&nbsp; &nbsp; is&nbsp;quantized and is&nbsp;given&nbsp;by&nbsp; &nbsp; E = h &nu;&nbsp; (1) &nbsp; &nbsp; where&nbsp;&nu;&nbsp;is&nbsp;the&nbsp;frequency&nbsp;associated&nbsp;with the&nbsp;radiation.&nbsp; Einstein&rsquo;s&nbsp;mass-energy&nbsp;relation states&nbsp;that&nbsp;&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;&nbsp; E = mc 2&nbsp; &nbsp; &#8230; <a title=\"Quantum Physics &#8211; RGPV Engineering Physics Notes\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/quantum-physics-from-rgpv-engineering-physics-notes\/\" aria-label=\"More on Quantum Physics &#8211; RGPV Engineering Physics Notes\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[2924],"tags":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/227058"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=227058"}],"version-history":[{"count":2,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/227058\/revisions"}],"predecessor-version":[{"id":227325,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/227058\/revisions\/227325"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=227058"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=227058"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=227058"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}