garmоnik оstsillyatоr, enеrgiya spеktri, хususiy funktsiyalari va matritsaviy elеmеntlari

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1424107532_60076.doc garmð¾nik ð¾stsillyatð¾r, enðµrgiya spðµktri, ñ ususiy funktsiyalari va matritsaviy elðµmðµntlari rðja: kirish: mumtð¾z fizikada ð¾stsillatð¾r prð¾blðµmasi asð¾siy qism: 1. garmonik ostilator masalasi 2. garmonik ostilatorning shryodingðµr tðµnglamasi buyicha ðµchilishi 3. garmonik ostilatorning matritsaviy hal etilishi 4. garmonik ostilator masalasida matritsaviy elðµmðµntlar, enðµrgiyasi ð¥ulð¾sa: garmonik ostilator masalasining fizikadagi rð¾li zarrachaning potentsial chuqurlik ichidagi harakati. erkin harakatlanayotgan mikrozarracha potentsial toâsiqdan qanday qaytishini oârgandik. potentsial toâsiq (e . potentsial chuqurlik tashqarisida esa. [image: image67.wmf]u e > (29) ni hisobga olib kvant ostilatori uchun shryodinger tenglamasini qoâyidagicha yozamiz: [image: image68.wmf]0 2 2 2 2 0 0 2 0 2 2 = ã· ã· ã¸ ã¶ ã§ ã§ ã¨ ã¦ - + y w y x m e m dx d h (30) bu tenglamani yechish uchun [image: image69.wmf]h 0 0 w h m x = (31) yangi uzgaruvchiga oâtamiz. u hð¾lda (29) tenglamaning koârinishi qoâyidagicha boâladi: [image: image70.wmf]( ) 0 2 2 2 …

qoâyidagi koârinishda izlaymiz [image: image75.wmf]( ) h j y 2 1 - = e (33) bu erdan koârinadiki [image: image76.wmf]â¥ â® h boâlganda [image: image77.wmf]0 â® y boâladi. (33) formulada [image: image78.wmf]y [image: image79.wmf]h uzgaruvchiga bð¾gâliq boâlgan yangi funktsiya .(33) ni (32) ga qoâysak [image: image80.wmf]y ga nisbatan yangi tenglama [image: image81.wmf]( ) 0 1 2 2 2 = - + - j c h j h h j d d d d (34) xosil boâladi. oxirgi tenglamaning ðµchimini qoâyidagi [image: image82.wmf]ã¥ = = 0 v v v b h j (35) koârinishda yozamiz. (35) koâp hadli (34) tenglamaning yechimi boâlishi uchun uni qanð¾atlantirish kerak. shuning uchun (35) ni (34) ga qoâysak qoâyidagi algebraik tenglama xosil boâladi: [image: image83.wmf]( ) ( ) [ ] 0 1 2 1 0 2 = - + - - ã¥ = - v v v v v v v b h c h (36) …

2 1 1 2 2 + = - + = + v v v b b v v c (38) shryodinger tenglamasining yechimi chekli boâlishi kerak. (35) qator esa hadlar sð¾ni ð¾rtishi bilan cheksiz ortib boraveradi. shuning uchun (35) qatorni qandaydir haddan boshlab uzish lozim. ana shu uzilayotgan hadning nomeri [image: image94.wmf]n v = bulsin. u hð¾lda (38) ni qoâyidagicha yozish mumkin [image: image95.wmf]( ) ( ) 2 1 1 2 2 + + - + = + n n n b b n n c (39) qator [image: image96.wmf]n -haddan boshlab uzilishi uchun [image: image97.wmf]0 â¹ n b ammð¾ [image: image98.wmf]0 2 = + n b boâlishi kerak. u hð¾lda [image: image99.wmf]4 + n b va undan keyingi barcha hadlar koeffitsientlari hamnulga teng boâladi. buning uchun (39) formulada [image: image100.wmf]0 1 2 = - + c n (40) boâlishi kerak. bu erga [image: image101.wmf]c ning qiymatini [image: image102.wmf]ã· ã· …

sak, sn = [image: image111.wmf]y n ð¡ (45) kðµlib chiqadi. u hð¾lda (42) ni umumiy hð¾lda qoâyidagicha yozish mumkin: [image: image112.wmf] embed equation.3 [image: image113.wmf]( ) 0 0 0 0 1 0 ; ! 2 1 2 0 w y m x x x h e x n n x n x x z n n h = ã· ã· ã¸ ã¶ ã§ ã§ ã¨ ã¦ = ã· ã· ã¸ ã¶ ã§ ã§ ã¨ ã¦ - (46) aniqlangan natijani tasavvur qilish uchun n ning bir kancha qiymatlariga mos kelgan xususiy energiyalar va xususiy funktsiyalar qiymatlarini keltiraylik: [image: image114.wmf]2 , 0 0 0 w h = = e n [image: image115.wmf]2 1 0 0 h y z e c - = ; [image: image116.wmf]0 1 2 3 , 1 w h = = e n , [image: image117.wmf]2 1 0 1 1 h y y z e z c - = …

n [image: image126.wmf]( ) 2 x d ni [image: image127.wmf]2 x bilan, [image: image128.wmf]( ) 2 x p ni [image: image129.wmf]2 p bilan almashtirish mumkin. u hð¾lda (49) munosabatdan [image: image130.wmf]2 2 2 4 x p x h = ni (48) ga qoâysak, [image: image131.wmf]2 8 2 2 0 0 2 0 x m x m e w + = h (50) kelib chiqadi. aniqlangan (50) natija [image: image132.wmf]2 x ning xech qanday qiymatida nolga aylanmaydi; [image: image133.wmf]0 2 â® x boâlsa, (50) da birinchi had [image: image134.wmf]â¥ ga intiladi, [image: image135.wmf]â¥ â® x boâlsa, ikkinchi had [image: image136.wmf]â¥ ga intiladi. endi (50) ning eng kichik qiymatini aniqlaylik. buning uchun (50) ni [image: image137.wmf]2 x boâyicha bir marta differentsiallab, chiqqan natijani nð¾lga tenglashtirib [image: image138.wmf]2 x ning qaysi qiymatida e minimum boâlishini aniqlaymiz. shu yoâl bilan topilgan [image: image139.wmf]2 x ning qiymatini (50) ga qoâysak, [image: image140.wmf]2 0 min w …

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1424107532_60076.doc garmð¾nik ð¾stsillyatð¾r, enðµrgiya spðµktri, ñ ususiy funktsiyalari va matritsaviy elðµmðµntlari rðja: kirish: mumtð¾z fizikada ð¾stsillatð¾r prð¾blðµmasi asð¾siy qism: 1. garmonik ostilator masalasi 2. garmonik ostilatorning shryodingðµr tðµnglamasi buyicha ðµchilishi 3. garmonik ostilatorning matritsaviy hal etilishi 4. garmonik ostilator masalasida matritsaviy elðµmðµntlar, enðµrgiyasi ð¥ulð¾sa: garmonik ostilator masalasining fizikadagi rð¾li zarrachaning potentsial chuqurlik ichidagi harakati. erkin harakatlanayotgan mikrozarracha potentsial toâsiqdan qanday qaytishini oârgandik. potentsial toâsiq (e . potentsial chuqurlik tashqarisida esa. [image: image67.wmf]u e > (29) ni hisobga olib kvant ostilatori uchun shryodinger tenglamasini qoâyidagicha yozamiz...

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