نظرية الإضطراب Perturbation Theory ... ج (2) !!!

[رجب مصطفى]

بسم الله الرحمن الرحيم
والصلاة والسلام على رسول الله
"محمد بن عبدالله"
وعلى آله وصحبه أجمعين
السلام عليكم ورحمة الله وبركاته
إخواني زوار وأعضاء ومشرفي المنتدى الكرام
حياكم الله وبياكم وجعل الفردوس مثواي ومثواكم

http://www.monsterup.com/upload/1222637946.gif


نظرية الإضطراب Perturbation Theory ... ج (1) !!!

قبل الدخول ... صلِّ على حبيبك المصطفى (صلى الله عليه وسلم) وعلى آله وصحبه أجمعين (http://hazemsakeek.com/vb/showthread.php?25538-Perturbation-Theory-%281%29-%21%21%21)

http://www.monsterup.com/upload/1222637946.gif


لعهدٍ قطعته على نفسي اليوم ... هذه في عُجالة ... التصحيح من الدرجة الثانية Second Order Correction في ...



نظرية الإضطراب Perturbation Theory


الغير معتمدة على الزمن Time-Independent ...

ملحوظة ... الشرح مأخوذ من كتاب Introductory Quantum Mechanics – Liboff ... ولكنه طبعاً ليس بهذا التفصيل الممل (الذي أعتبره من حقوق الملكية الخاصة بي !!!) ...

*** لإيجاد التصحيح من الدرجة الثانية للدالة الموجية والطاقة (http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cphi_%7Bn%7D,E _%7Bn%7D)، (http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cphi_%7Bn%7D,E _%7Bn%7D%29%D8%8C) ينبغي حل المعادلة التالية (والتي هي معامل http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Clambda%5E%7B2 %7D):


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cleft&space;%28&space;%5Ch at%7BH%7D_%7B0%7D-E_%7Bn%7D%5E%7B%280%29%7D&space;%5Cright&space; %29%5Cphi_%7Bn%7D%5E%7B%282%29%7D=% 5Cleft&space;%28&space;E_%7Bn%7D%5E%7B%281%29%7 D&space;-%5Chat%7B%7BH%7D%27%7D%5Cright&space;%29% 5Cphi_%7Bn%7D%5E%7B%281%29%7D+E_%7B n%7D%5E%7B%282%29%7D%5Cphi_%7Bn%7D% 5E%7B%280%29%7D

بالفك والتوزيع واستخدام صورة الـ ket vectors نجد أن:



http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Chat%7BH%7D_%7 B0%7D%5Cleft&space;%7C%5Cphi_%7Bn%7D%5E%7 B%282%29%7D%5Cright&space;%5Crangle-E_%7Bn%7D%5E%7B%280%29%7D%5Cleft&space;%7 C%5Cphi_%7Bn%7D%5E%7B%282%29%7D%5Cr ight&space;%5Crangle=E_%7Bn%7D%5E%7B%281% 29%7D&space;%5Cleft&space;%7C%5Cphi_%7Bn%7D%5E% 7B%281%29%7D%5Cright&space;%5Crangle-%5Chat%7B%7BH%7D%27%7D%5Cleft&space;%7C%5 Cphi_%7Bn%7D%5E%7B%281%29%7D%5Crigh t&space;%5Crangle+E_%7Bn%7D%5E%7B%282%29% 7D%5Cleft&space;%7C%5Cphi_%7Bn%7D%5E%7B%2 80%29%7D%5Cright&space;%5Crangle


وحيث أن الهاملتونيان http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Chat%7BH%7D_%7 B0%7D يعمل فقط على الدالة http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cphi_%7Bn%7D%5 E%7B%282%29%7D وبالتالي يمكن أن ننشر أو نوزع expand هذه الدالة في صورة الدوال المميزة للهاميلتونيان السابق، أي أن:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cleft&space;%7C%5Cph i_%7Bn%7D%5E%7B%282%29%7D%5Cright&space;% 5Crangle&space;=&space;%5Csum_%7Bi%7DD_%7Bni%7D %5Cleft&space;%7C%5Cphi_%7Bi%7D%5E%7B%280 %29%7D%5Cright&space;%5Crangle

بالتعويض:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Csum_%7Bi%7DD_ %7Bni%7D%7BH%7D_%7B0%7D%5Cleft&space;%7C% 5Cphi_%7Bi%7D%5E%7B%280%29%7D%5Crig ht&space;%5Crangle-%5Csum_%7Bi%7DD_%7Bni%7DE_%7Bn%7D%5 E%7B%280%29%7D%5Cleft&space;%7C%5Cphi_%7B i%7D%5E%7B%280%29%7D%5Cright&space;%5Cran gle=E_%7Bn%7D%5E%7B%281%29%7D&space;%5Cle ft&space;%7C%5Cphi_%7Bn%7D%5E%7B%281%29%7 D%5Cright&space;%5Crangle-%5Chat%7B%7BH%7D%27%7D%5Cleft&space;%7C%5 Cphi_%7Bn%7D%5E%7B%281%29%7D%5Crigh t&space;%5Crangle+E_%7Bn%7D%5E%7B%282%29% 7D%5Cleft&space;%7C%5Cphi_%7Bn%7D%5E%7B%2 80%29%7D%5Cright&space;%5Crangle

بضرب طرفي المعادلة السابقة بالـ bra vector للدالة المميزة ولكن في حالة أخرى، أي:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cleft&space;%5Clangl e&space;%5Cphi_%7Bj%7D%5E%7B%280%29%7D%5C right&space;%7C

نجد أن:



http://latex.codecogs.com/gif.latex?%5Csum_%7Bi%7DD_%7Bni%7D% 5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E% 7B%280%29%7D%5Cright&space;%7C%7BH%7D_%7B 0%7D%5Cleft&space;%7C%5Cphi_%7Bi%7D%5E%7B %280%29%7D%5Cright&space;%5Crangle-%5Csum_%7Bi%7DD_%7Bni%7D%5Cleft&space;%5C langle&space;%5Cphi_%7Bj%7D%5E%7B%280%29% 7D%5Cright&space;%7CE_%7Bn%7D%5E%7B%280%2 9%7D%5Cleft&space;%7C%5Cphi_%7Bi%7D%5E%7B %280%29%7D%5Cright&space;%5Crangle&space;=&space;E_%7 Bn%7D%5E%7B%281%29%7D%5Cleft&space;%5Clan gle&space;%5Cphi_%7Bj%7D%5E%7B%280%29%7D% 5Cright&space;%5Cleft&space;%7C%5Cphi_%7Bn%7D%5 E%7B%281%29%7D%5Cright&space;%5Crangle&space;-&space;%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5 E%7B%280%29%7D%5Cright&space;%7C&space;%5Cleft&space; %5Chat%7B%7BH%7D%27%7D%5Cleft&space;%7C%5 Cphi_%7Bn%7D%5E%7B%281%29%7D%5Crigh t&space;%5Crangle&space;+&space;E_%7Bn%7D%5E%7B%282%2 9%7D%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7 D%5E%7B%280%29%7D%5Cright&space;%5Cleft&space;% 7C&space;%5Cleft&space;%5Cphi_%7Bn%7D%5E%7B%280 %29%7D&space;%5Cright&space;%5Crangle&space;%5Cright

وومنها نحصل على:


http://latex.codecogs.com/gif.latex?%5Csum_%7Bi%7DD_%7Bni%7DE _%7Bi%7D%5E%7B%280%29%7D%5Cleft&space;%5C langle&space;%5Cphi_%7Bj%7D%5E%7B%280%29% 7D%5Cright&space;%5Cleft&space;%7C%5Cphi_%7Bi%7 D%5E%7B%280%29%7D%5Cright&space;%5Crangle-%5Csum_%7Bi%7DD_%7Bni%7DE_%7Bn%7D%5 E%7B%280%29%7D%5Cleft&space;%5Clangle&space;%5C phi_%7Bj%7D%5E%7B%280%29%7D%5Cright &space;%5Cleft&space;%7C%5Cphi_%7Bi%7D%5E%7B%28 0%29%7D%5Cright&space;%5Crangle=E_%7Bn%7D %5E%7B%281%29%7D&space;%5Cleft&space;%5Clangle&space; %5Cphi_%7Bj%7D%5E%7B%280%29%7D%5Cle ft&space;%7C%5Cphi_%7Bn%7D%5E%7B%281%29%7 D%5Cright&space;%5Crangle-%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E %7B%280%29%7D%5Cleft&space;%7C%5Chat%7B%7 BH%7D%27%7D%5Cleft&space;%7C%5Cphi_%7Bn%7 D%5E%7B%281%29%7D%5Cright&space;%5Crangle +E_%7Bn%7D%5E%7B%282%29%7D%5Cleft&space;% 5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E% 7B%280%29%7D%5Cleft&space;%7C%5Cphi_%7Bn% 7D%5E%7B%280%29%7D%5Cright%5Cright&space; %5Crangle

حيث:


http://latex.codecogs.com/gif.latex?%5Cleft&space;%5Clangle&space;%5Cphi_ %7Bj%7D%5E%7B%280%29%7D%5Cright&space;%7C %7BH%7D_%7B0%7D%5Cleft&space;%7C%5Cphi_%7 Bi%7D%5E%7B%280%29%7D%5Cright&space;%5Cra ngle=E_%7Bi%7D%5E%7B%280%29%7D%5Cle ft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E%7B%2 80%29%7D%5Cright&space;%5Cleft&space;%7C%5Cphi_ %7Bi%7D%5E%7B%280%29%7D%5Cright&space;%5C rangle

والآن بفرض أن http://latex.codecogs.com/gif.latex?i=j، (http://latex.codecogs.com/gif.latex?i=j%D8%8C) فإن:


http://latex.codecogs.com/gif.latex?%5Csum_%7Bj%7DD_%7Bnj%7DE _%7Bj%7D%5E%7B%280%29%7D%5Cleft&space;%5C langle&space;%5Cphi_%7Bj%7D%5E%7B%280%29% 7D%5Cright&space;%5Cleft&space;%7C%5Cphi_%7Bj%7 D%5E%7B%280%29%7D%5Cright&space;%5Crangle-%5Csum_%7Bj%7DD_%7Bnj%7DE_%7Bn%7D%5 E%7B%280%29%7D%5Cleft&space;%5Clangle&space;%5C phi_%7Bj%7D%5E%7B%280%29%7D%5Cright &space;%5Cleft&space;%7C%5Cphi_%7Bj%7D%5E%7B%28 0%29%7D%5Cright&space;%5Crangle=E_%7Bn%7D %5E%7B%281%29%7D&space;%5Cleft&space;%5Clangle&space; %5Cphi_%7Bj%7D%5E%7B%280%29%7D%5Cle ft&space;%7C%5Cphi_%7Bn%7D%5E%7B%281%29%7 D%5Cright&space;%5Crangle-%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E %7B%280%29%7D%5Cleft&space;%7C%5Chat%7B%7 BH%7D%27%7D%5Cleft&space;%7C%5Cphi_%7Bn%7 D%5E%7B%281%29%7D%5Cright&space;%5Crangle +E_%7Bn%7D%5E%7B%282%29%7D%5Cleft&space;% 5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E% 7B%280%29%7D%5Cleft&space;%7C%5Cphi_%7Bn% 7D%5E%7B%280%29%7D%5Cright%5Cright&space; %5Crangle

ومنها نحصل على:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%7B%5Ccolor%7Bbl ue%7D&space;%5Csum_%7Bj%7DD_%7Bnj%7D%5Cle ft&space;%28E_%7Bj%7D%5E%7B%280%29%7D-E_%7Bn%7D%5E%7B%280%29%7D&space;%5Cright&space; %29=E_%7Bn%7D%5E%7B%281%29%7D&space;%5Cle ft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E%7B%2 80%29%7D%7C&space;%5Cphi_%7Bn%7D%5E%7B%28 1%29%7D%5Cright&space;%5Crangle-%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E %7B%280%29%7D%7C%5Chat%7B%7BH%7D%27 %7D%7C%5Cphi_%7Bn%7D%5E%7B%281%29%7 D%5Cright&space;%5Crangle+E_%7Bn%7D%5E%7B %282%29%7D%5Cdelta&space;_%7Bjn%7D%5Crigh t&space;%5Crangle%7D

وهذه هي المعادلة المحورية للقادم ...

[رجب مصطفى]

*** تابع ...

عند http://latex.codecogs.com/gif.latex?j=n، (http://latex.codecogs.com/gif.latex?j=n%D8%8C) فإن:


http://latex.codecogs.com/gif.latex?E_%7Bn%7D%5E%7B%282%29%7D =%5Cleft&space;%5Clangle&space;%5Cphi_%7Bn%7D%5 E%7B%280%29%7D%7C%5Chat%7B%7BH%7D%2 7%7D%7C%5Cphi_%7Bn%7D%5E%7B%281%29% 7D%5Cright&space;%5Crangle%5Cright&space;%5Cran gle

حيث:


http://latex.codecogs.com/gif.latex?%5Cleft&space;%5Clangle&space;%5Cphi_ %7Bn%7D%5E%7B%280%29%7D%7C&space;%5Cphi_% 7Bn%7D%5E%7B%281%29%7D%5Cright&space;%5Cr angle=0

لكن:



http://latex.codecogs.com/gif.latex?%5Cphi_%7Bn%7D%5E%7B%281% 29%7D=%5Csum_%7Bi%5Cneq&space;n%7D%5Cfrac %7B%5Chat%7B%7BH%7D%27%7D_%7Bin%7D% 7D%7BE_%7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D%7D%5Cphi_ %7Bi%7D%5E%7B%280%29%7D

إذاً ...


http://latex.codecogs.com/gif.latex?E_%7Bn%7D%5E%7B%282%29%7D =%5Cleft&space;%5Clangle&space;%5Cphi_%7Bn%7D%5 E%7B%280%29%7D%7C%5Chat%7B%7BH%7D%2 7%7D%7C%5Cphi_%7Bn%7D%5E%7B%281%29% 7D%5Cright&space;%5Crangle%5Cright&space;%5Cran gle=&space;%5Csum_%7Bi%5Cneq&space;n%7D%5Cleft&space; %5Clangle&space;%5Cphi_%7Bn%7D%5E%7B%280% 29%7D%7C%5Cfrac%7B%5Chat%7B%7BH%7D% 27%7D&space;%5Chat%7B%7BH%7D%27%7D_%7Bin% 7D%7D%7BE_%7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D%7D%7C%5Cp hi_%7Bi%7D%5E%7B%280%29%7D%5Cright&space; %5Crangle%5Cright&space;%5Crangle

ولكن


http://latex.codecogs.com/gif.latex?%5Chat%7B%7BH%7D%27%7D_%7 Bni%7D=%5Cleft&space;%5Clangle&space;%5Cphi_%7B n%7D%5E%7B%280%29%7D%7C%5Chat%7B%7B H%7D%27%7D%7C%5Cphi_%7Bi%7D%5E%7B%2 80%29%7D%5Cright&space;%5Crangle%5Cright&space; %5Crangle

إذاً ...


http://latex.codecogs.com/gif.latex?E_%7Bn%7D%5E%7B%282%29%7D =&space;%5Csum_%7Bi%5Cneq&space;n%7D%5Cfrac%7B% 5Chat%7B%7BH%7D%27%7D_%7Bni%7D&space;%5Ch at%7B%7BH%7D%27%7D_%7Bin%7D%7D%7BE_ %7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D%7D

وفي النهاية نجد أن التصحيح الثاني للطاقة هو ...


http://latex.codecogs.com/gif.latex?%5Clarge&space;%7B%5Ccolor%7Bre d%7D&space;E_%7Bn%7D%5E%7B%282%29%7D=%5Cs um_%7Bi%5Cneq&space;n%7D%5Cfrac%7B%5Cleft &space;%7C%5Chat%7B%7BH%7D%27%7D_%7Bin%7D &space;%5Cright&space;%7C%5E%7B2%7D%7D%7BE_%7Bn %7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D%7D%7D

وعليها تكون المعادلة الكلية للطاقة هي


http://latex.codecogs.com/gif.latex?%5Clarge&space;%7B%5Ccolor%7Bbl ue%7D&space;E_%7Bn%7D=E_%7Bn%7D%5E%7B%280 %29%7D+%5Chat%7B%7BH%7D%27%7D_%7Bnn %7D+%5Csum_%7Bi%5Cneq&space;n%7D%5Cfrac%7 B%5Cleft&space;%7C%5Chat%7B%7BH%7D%27%7D_ %7Bin%7D&space;%5Cright&space;%7C%5E%7B2%7D%7D% 7BE_%7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D%7D%7D

[رجب مصطفى]

*** والآن ... لإيجاد التصحيح من الدرجة الثانية للدالة الموجية، نعود للمعادلة المحورية ذات اللون الأزرق (فقط لتميزها)، وبفرض الحالة التي فيها http://latex.codecogs.com/gif.latex?j&space;%5Cneq&space;n، (http://latex.codecogs.com/gif.latex?j&space;%5Cneq&space;n%D8%8C) فإن:


http://latex.codecogs.com/gif.latex?E_%7Bn%7D%5E%7B%281%29%7D &space;%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5 E%7B%280%29%7D%7C&space;%5Cphi_%7Bn%7D%5E %7B%281%29%7D%5Cright&space;%5Crangle-%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E %7B%280%29%7D%7C%5Chat%7B%7BH%7D%27 %7D%7C%5Cphi_%7Bn%7D%5E%7B%281%29%7 D%5Cright&space;%5Crangle=%5Csum_%7Bj%7DD _%7Bnj%7D%5Cleft&space;%28E_%7Bj%7D%5E%7B %280%29%7D-E_%7Bn%7D%5E%7B%280%29%7D&space;%5Cright&space; %29

وبالضرب في – 1 ... (أي ترتيب فقط) ...


http://latex.codecogs.com/gif.latex?%5Cleft&space;%5Clangle&space;%5Cphi_ %7Bj%7D%5E%7B%280%29%7D%7C%5Chat%7B %7BH%7D%27%7D%7C%5Cphi_%7Bn%7D%5E%7 B%281%29%7D%5Cright&space;%5Crangle-E_%7Bn%7D%5E%7B%281%29%7D&space;%5Cleft&space;% 5Clangle&space;%5Cphi_%7Bj%7D%5E%7B%280%2 9%7D%7C&space;%5Cphi_%7Bn%7D%5E%7B%281%29 %7D%5Cright&space;%5Crangle=%5Csum_%7Bj%7 DD_%7Bnj%7D%5Cleft&space;%28E_%7Bn%7D%5E% 7B%280%29%7D-E_%7Bj%7D%5E%7B%280%29%7D&space;%5Cright&space; %29

ولكن من الممكن أن نكتب ...


http://latex.codecogs.com/gif.latex?%5Cphi_%7Bn%7D%5E%7B%281% 29%7D=%5Csum_%7Bk%5Cneq&space;n%7D%5Cfrac %7B%5Chat%7B%7BH%7D%27%7D_%7Bkn%7D% 7D%7BE_%7Bn%7D%5E%7B%280%29%7D-E_%7Bk%7D%5E%7B%280%29%7D%7D%5Cphi_ %7Bk%7D%5E%7B%280%29%7D

وأيضاً ...


http://latex.codecogs.com/gif.latex?E_%7Bn%7D%5E%7B%281%29%7D =%5Chat%7B%7BH%7D%27%7D_%7Bnn%7D

بالتعويض ...


http://latex.codecogs.com/gif.latex?%5Cleft&space;%5Clangle&space;%5Cphi_ %7Bj%7D%5E%7B%280%29%7D%7C%5Csum_%7 Bk%5Cneq&space;n%7D%5Cfrac%7B%5Chat%7B%7B H%7D%27%7D%5Chat%7B%7BH%7D%27%7D_%7 Bkn%7D%7D%7BE_%7Bn%7D%5E%7B%280%29% 7D-E_%7Bk%7D%5E%7B%280%29%7D%7D%7C%5Cp hi_%7Bk%7D%5E%7B%280%29%7D%5Cright&space; %5Crangle-%5Chat%7B%7BH%7D%27%7D_%7Bnn%7D&space;%5C left&space;%5Clangle&space;%5Cphi_%7Bj%7D%5E%7B %280%29%7D%7C%5Csum_%7Bk%5Cneq&space;n%7D %5Cfrac%7B%5Chat%7B%7BH%7D%27%7D_%7 Bkn%7D%7D%7BE_%7Bn%7D%5E%7B%280%29% 7D-E_%7Bk%7D%5E%7B%280%29%7D%7D%7C%5Cp hi_%7Bk%7D%5E%7B%280%29%7D%5Cright% 5Crangle=%5Csum_%7Bj%7DD_%7Bnj%7D%5 Cleft&space;%28E_%7Bn%7D%5E%7B%280%29%7D-E_%7Bj%7D%5E%7B%280%29%7D&space;%5Cright&space; %29

وكما فعلنا سابقاً ... نجد أن:


http://latex.codecogs.com/gif.latex?%5Chat%7B%7BH%7D%27%7D_%7 Bjk%7D=%5Cleft&space;%5Clangle&space;%5Cphi_%7B j%7D%5E%7B%280%29%7D%7C%5Chat%7B%7B H%7D%27%7D%7C%5Cphi_%7Bk%7D%5E%7B%2 80%29%7D%5Cright&space;%5Crangle

وعليه فإن:


http://latex.codecogs.com/gif.latex?%5Chat%7B%7BH%7D%27%7D_%7 Bkn%7D=%5Cleft&space;%5Clangle&space;%5Cphi_%7B k%7D%5E%7B%280%29%7D%7C%5Chat%7B%7B H%7D%27%7D%7C%5Cphi_%7Bn%7D%5E%7B%2 80%29%7D%5Cright&space;%5Crangle

إذاً ...


http://latex.codecogs.com/gif.latex?%5Cleft&space;%5Clangle&space;%5Cphi_ %7Bj%7D%5E%7B%280%29%7D%7C%5Chat%7B %7BH%7D%27%7D_%7Bkn%7D%7C%5Cphi_%7B k%7D%5E%7B%280%29%7D%5Cright&space;%5Cran gle=%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj%7 D%5E%7B%280%29%7D%7C%5Cleft&space;%5Clang le&space;%7B%5Ccolor%7Bred%7D&space;%5Cphi_%7Bk %7D%5E%7B%280%29%7D%7D%7C%5Chat%7B% 7BH%7D%27%7D%7C%5Cphi_%7Bn%7D%5E%7B %280%29%7D%5Cright&space;%5Crangle&space;%7C%7B %5Ccolor%7Bred%7D&space;%5C,&space;%5Cphi_%7Bk% 7D%5E%7B%280%29%7D%7D%5Cright&space;%5Cra ngle=%5Cleft&space;%5Clangle&space;%5Cphi_%7Bj% 7D%5E%7B%280%29%7D%7C%5Cleft&space;%5Chat %7B%7BH%7D%27%7D%7C%5Cphi_%7Bn%7D%5 E%7B%280%29%7D%5Cright&space;%5Crangle=%7 B%5Ccolor%7Bblue%7D&space;%5Chat%7B%7BH%7 D%27%7D_%7Bjn%7D%7D

[رجب مصطفى]

ومنها يمكن أن نضع


http://latex.codecogs.com/gif.latex?k=j

في هذا التعبير "فقط" !!!...

والآن ...


http://latex.codecogs.com/gif.latex?%5Csum_%7Bk%5Cneq&space;n%7D%5C frac%7B%5Chat%7B%7BH%7D%27%7D_%7Bjk %7D%5Chat%7B%7BH%7D%27%7D_%7Bkn%7D% 7D%7BE_%7Bn%7D%5E%7B%280%29%7D-E_%7Bk%7D%5E%7B%280%29%7D%7D-%5Csum_%7Bj%5Cneq&space;n%7D%5Cfrac%7B%5C hat%7B%7BH%7D%27%7D_%7Bnn%7D%5Chat% 7B%7BH%7D%27%7D_%7Bjn%7D%7D%7BE_%7B n%7D%5E%7B%280%29%7D-E_%7Bj%7D%5E%7B%280%29%7D%7D=%5Csum _%7Bj%5Cneq&space;n%7DD_%7Bnj%7D%5Cleft&space;% 28E_%7Bn%7D%5E%7B%280%29%7D-E_%7Bj%7D%5E%7B%280%29%7D&space;%5Cright&space; %29

أو:


http://latex.codecogs.com/gif.latex?D_%7Bnj%7D=%5Csum_%7Bj%5C neq&space;n%7D%5Cfrac%7B1%7D%7B%28E_%7Bn% 7D%5E%7B%280%29%7D-E_%7Bj%7D%5E%7B%280%29%7D%29%7D%5Cs um_%7Bk%5Cneq&space;n%7D%5Cfrac%7B%5Chat% 7B%7BH%7D%27%7D_%7Bjk%7D%5Chat%7B%7 BH%7D%27%7D_%7Bkn%7D%7D%7BE_%7Bn%7D %5E%7B%280%29%7D-E_%7Bk%7D%5E%7B%280%29%7D%7D-%5Csum_%7Bj%5Cneq&space;n%7D%5Cfrac%7B%5C hat%7B%7BH%7D%27%7D_%7Bnn%7D%5Chat% 7B%7BH%7D%27%7D_%7Bjn%7D%7D%7B%5Cle ft&space;%28E_%7Bn%7D%5E%7B%280%29%7D-E_%7Bj%7D%5E%7B%280%29%7D&space;%5Cright&space; %29%5E%7B2%7D%7D

وكما فرضنا في البداية أن:


http://latex.codecogs.com/gif.latex?%5Cphi_%7Bn%7D%5E%7B%282% 29%7D=%5Csum_%7Bi%7DD_%7Bni%7D%5Cph i_%7Bi%7D%5E%7B%280%29%7D

لذا بوضع http://latex.codecogs.com/gif.latex?j=i، (http://latex.codecogs.com/gif.latex?j=i%D8%8C) نجد أن:


http://latex.codecogs.com/gif.latex?%5Csum_%7Bi%7DD_%7Bni%7D= %5Csum_%7Bi%5Cneq&space;n%7D%5Cfrac%7B1%7 D%7B%28E_%7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D%29%7D%5Cs um_%7Bk%5Cneq&space;n%7D%5Cfrac%7B%5Chat% 7B%7BH%7D%27%7D_%7Bjk%7D%5Chat%7B%7 BH%7D%27%7D_%7Bkn%7D%7D%7BE_%7Bn%7D %5E%7B%280%29%7D-E_%7Bk%7D%5E%7B%280%29%7D%7D-%5Csum_%7Bi%5Cneq&space;n%7D%5Cfrac%7B%5C hat%7B%7BH%7D%27%7D_%7Bnn%7D%5Chat% 7B%7BH%7D%27%7D_%7Bin%7D%7D%7B%5Cle ft&space;%28E_%7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D&space;%5Cright&space; %29%5E%7B2%7D%7D

وفي النهاية نجد أن التصحيح الثاني للدالة الموجية هو ...


http://latex.codecogs.com/gif.latex?%5Cphi_%7Bn%7D%5E%7B%282% 29%7D=%5Cleft&space;%28%5Csum_%7Bi%5Cneq&space; n%7D%5Cfrac%7B1%7D%7B%28E_%7Bn%7D%5 E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D%29%7D%5Cs um_%7Bk%5Cneq&space;n%7D%5Cfrac%7B%5Chat% 7B%7BH%7D%27%7D_%7Bjk%7D%5Chat%7B%7 BH%7D%27%7D_%7Bkn%7D%7D%7BE_%7Bn%7D %5E%7B%280%29%7D-E_%7Bk%7D%5E%7B%280%29%7D%7D-%5Csum_%7Bi%5Cneq&space;n%7D%5Cfrac%7B%5C hat%7B%7BH%7D%27%7D_%7Bnn%7D%5Chat% 7B%7BH%7D%27%7D_%7Bin%7D%7D%7B%5Cle ft&space;%28E_%7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D&space;%5Cright&space; %29%5E%7B2%7D%7D&space;%5Cright&space;%29%5Cphi _%7Bi%7D%5E%7B%280%29%7D

وعليها تكون المعادلة الكلية للدالة الموجية هي


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Clarge&space;%7B%5Cc olor%7Bblue%7D&space;%5Cphi_%7Bn%7D=%5Cph i_%7Bn%7D%5E%7B%280%29%7D+%5Csum_%7 Bi%5Cneq&space;n%7D%5Cleft&space;%28%5Cfrac%7B% 5Chat%7B%7BH%7D%27%7D_%7Bin%7D%7D%7 BE_%7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D%7D-%5Cfrac%7B%5Chat%7B%7BH%7D%27%7D_%7 Bnn%7D%5Chat%7B%7BH%7D%27%7D_%7Bin% 7D%7D%7B%5Cleft&space;%28E_%7Bn%7D%5E%7B% 280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D&space;%5Cright&space; %29%5E%7B2%7D%7D+%5Cleft&space;%5Csum_%7B k%5Cneq&space;n%7D%5Cfrac%7B%5Chat%7B%7BH %7D%27%7D_%7Bjk%7D%5Chat%7B%7BH%7D% 27%7D_%7Bkn%7D%7D%7B%5Cleft&space;%28&space;E_% 7Bn%7D%5E%7B%280%29%7D-E_%7Bi%7D%5E%7B%280%29%7D&space;%5Cright&space; %29%5Cleft&space;%28E_%7Bn%7D%5E%7B%280%2 9%7D-E_%7Bk%7D%5E%7B%280%29%7D&space;%5Cright&space; %29%7D&space;%5Cright&space;&space;%5Cright&space;%29%5Cphi _%7Bi%7D%5E%7B%280%29%7D%7D



*** *** ***

وآخر دعوانا أن الحمد لله رب العالمين
وصلي اللهم وبارك على نبي الرحمة سيد ولد أدم أبى القاسم "محمد بن عبد الله" وعلى آله الطيبين الطاهرين وعلى أصحابه الغر الميامين وسلم تسليماً كثيراً
لا تنسونا من صالح دعائكم

[شعاع الفيزيا]

السلام عليكم الف شكر وان شاء الله يستفيدوا منه الاعضاء وجعله في ميزان جسناتك يوم الحساب وصلي عل حبيبك محمد ابن عبدالله

[deadheart]

ايه الابداع ده كلو هوا انتا كنت مخبي كل ده فين ممكن اعرف

جزاك الله خيراً استاذي سامحنا على التقصير

اثابكم الله به الجنة وحرم وجهكم على النار يوم القيامة

[غـــرور]

جزاك الله خـــــــير بوركت

[بنت الوفء]

الى الاخ الفاضل/ رجب مصطفى
أود ان اشكرك على المجهودالفوق ممتاز في مساعدتي ومساعدة الاخرين في اثبات نظرية الاضطراب نفعك الله به ونفع غيرك به وجعله في ميزان حساناتك
لا أدري كيف أشكرك على هذه المساعده
والتي لاتقدر بثمن
فألف مليار شكر
على جهدك فى سبيل امتك
فشكرا من أعماق قلبي ياشمعة المنتدى

[محمد مصطفى]

انا مقبل على دراسة هذه النظريه فى المحاضرات القادمه فى مقرر ضوء كمى quantum optics

شكرا لك اخى العزيز رجب

ولو سمحت ياريت اى نصائح للتعامل مع هذه المعادلات الكثيره جدا لان فى مقرر ضوء كمى ناخد العديد من المواضيع والتى تحتوى ع كم هائل من المعادلات

السؤال بطريقه اخى كيف كنت تذاكر مثل هذه المقررات اخى رجب ؟؟؟

شكرا على مجهودك المتميز

تحياتى

[محبة الرسول]

جزاك الله خيرا على هذا المجهود....أسأل الله ان ينفع بك امة الاسلام...

[spring]

شكرا جزيلا وبارك الله فيك