رجب مصطفى
05-04-2010, 12:49 AM
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 ...
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:
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حيث:
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والآن بفرض أن http://latex.codecogs.com/gif.latex?i=j، (http://latex.codecogs.com/gif.latex?i=j%D8%8C) فإن:
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ومنها نحصل على:
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وهذه هي المعادلة المحورية للقادم ...