التطور الزمني للقيم المتوقعة ...

[رجب مصطفى]

بسم الله الرحمن الرحيم

الحمد لله الذي صدق وعده، ونصر عبده، وأعز جنده، وهزم الأحزاب وحده، والصلاة السلام على من لا نبي بعده، رسوله الذي هدى به الأنام، وكشف به شبهات الأوهام، وعلى آله الطيبين الأطهار، وأصحابه المجاهدين الأبرار، الذين أغاظ الله بهم الكفار، وبسط بهم رحمته في جميع الأقطار


أما بعد:

*** ملحوظة هامة جداً كالعادة ... هذا الموضوع حصري لـ "منتدى الفيزياء التعليمي" فقط، غير ذلك سيكون واضعه سارقاً له !!!


التطور الزمني للقيم المتوقعة

Time Development of Expectation Values


من المعروف أن القيمة المتوقعة expectation value لخاصيةٍ ما، ولتكن http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Chat%7BA%7D ، تُعطى من العلاقة:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cleft&space;%5Clangl e&space;%5Chat%7BA%7D&space;%5Cright&space;%5Crangle= %5Cint%5Cpsi%5E%7B%5Cast%7D%5C;&space;%5C hat%7BA%7D&space;%5Cpsi%5C;&space;dx


ولكن ... دعونا نرى سلوك هذه القيمة مع الزمن، وذلك من خلال تفاضل هذه العلاقة بالنسبة للزمن:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7Bd%7D%7 Bdt%7D%5Cleft&space;%5Clangle&space;%5Chat%7BA% 7D&space;%5Cright&space;%5Crangle=%5Cfrac%7Bd%7 D%7Bdt%7D%5Cint%5Cpsi%5E%7B%5Cast%7 D%5C;&space;%5Chat%7BA%7D&space;%5Cpsi%5C;&space;dx=% 5Cint%5Cfrac%7B%5Cpartial&space;%7D%7B%5C partial&space;t%7D%5Cleft&space;%28%5Cpsi%5E%7B %5Cast%7D%5C;&space;%5Chat%7BA%7D&space;%5Cpsi&space; %5Cright&space;%29%5C;&space;dx


حيث:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7B%5Cpar tial&space;%7D%7B%5Cpartial&space;t%7D%5Cleft&space;% 28%5Cpsi%5E%7B%5Cast%7D%5C;&space;%5Chat% 7BA%7D&space;%5Cpsi&space;%5Cright&space;%29=%5Cfrac% 7B%5Cpartial&space;%5Cpsi%5E%7B%5Cast%7D% 7D%7B%5Cpartial&space;t%7D%5Cleft&space;%28%5Ch at%7BA%7D&space;%5Cpsi&space;%5Cright&space;%29+%5Cps i%5E%7B%5Cast%7D%5Cfrac%7B%5Cpartia l&space;%7D%7B%5Cpartial&space;t%7D%5Cleft&space;%28% 5Chat%7BA%7D&space;%5Cpsi&space;%5Cright&space;%29


ولكن:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7B%5Cpar tial&space;%7D%7B%5Cpartial&space;t%7D%5Cleft&space;% 28%5Chat%7BA%7D&space;%5Cpsi&space;%5Cright&space;%29 =%5Cfrac%7B%5Cpartial&space;%5Chat%7BA%7D %7D%7B%5Cpartial&space;t%7D%5Cleft&space;%28&space;%5 Cpsi&space;%5Cright&space;%29+%5Chat%7BA%7D&space;%5C ;%5Cfrac%7B%5Cpartial%5Cpsi&space;%7D%7B% 5Cpartial&space;t%7D


إذاً ...


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7B%5Cpar tial&space;%7D%7B%5Cpartial&space;t%7D%5Cleft&space;% 28%5Cpsi%5E%7B%5Cast%7D%5C;&space;%5Chat% 7BA%7D&space;%5Cpsi&space;%5Cright&space;%29=%5Cfrac% 7B%5Cpartial&space;%5Cpsi%5E%7B%5Cast%7D% 7D%7B%5Cpartial&space;t%7D%5Cleft&space;%28%5Ch at%7BA%7D&space;%5Cpsi&space;%5Cright&space;%29+%5Cps i%5E%7B%5Cast%7D%5Cleft&space;%28%5Cfrac% 7B%5Cpartial&space;%5Chat%7BA%7D%7D%7B%5C partial&space;t%7D%5Cleft&space;%28&space;%5Cpsi&space;%5Cr ight&space;%29+%5Chat%7BA%7D&space;%5C;%5Cfrac% 7B%5Cpartial%5Cpsi&space;%7D%7B%5Cpartial &space;t%7D&space;%5Cright&space;%29


ولكن، أيضاً ...


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Chat%7BH%7D%5C psi=i%5Chbar%5Cfrac%7B%5Cpartial%5C psi&space;%7D%7B%5Cpartial&space;t%7D%5Cqquad&space;% 5CRightarrow&space;%5Cqquad&space;%5Cfrac%7B%5C partial%5Cpsi&space;%7D%7B%5Cpartial&space;t%7D =-%5Cfrac%7Bi%7D%7B%5Chbar%7D%5Chat%7 BH%7D%5Cpsi&space;%5Cqquad,%5Cfrac%7B%5Cp artial%5Cpsi%5E%7B%5Cast%7D&space;%7D%7B% 5Cpartial&space;t%7D=%5Cfrac%7Bi%7D%7B%5C hbar%7D%5Chat%7BH%7D%5Cpsi%5E%7B%5C ast%7D


إذاً ...


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7B%5Cpar tial&space;%7D%7B%5Cpartial&space;t%7D%5Cleft&space;% 28%5Cpsi%5E%7B%5Cast%7D%5C;&space;%5Chat% 7BA%7D&space;%5Cpsi&space;%5Cright&space;%29=%5Cfrac% 7Bi%7D%7B%5Chbar%7D%5Chat%7BH%7D%5C psi%5E%7B%5Cast%7D%5Chat%7BA%7D&space;%5C psi+%5Cpsi%5E%7B%5Cast%7D%5Cfrac%7B %5Cpartial&space;%5Chat%7BA%7D%7D%7B%5Cpa rtial&space;t%7D%5Cleft&space;%28&space;%5Cpsi&space;%5Crig ht&space;%29-%5Cfrac%7Bi%7D%7B%5Chbar%7D%5Cpsi%5 E%7B%5Cast%7D%5Chat%7BA%7D&space;%5Chat%7 BH%7D%5Cpsi&space;%5Cright&space;


أو:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7B%5Cpar tial&space;%7D%7B%5Cpartial&space;t%7D%5Cleft&space;% 28%5Cpsi%5E%7B%5Cast%7D%5C;&space;%5Chat% 7BA%7D&space;%5Cpsi&space;%5Cright&space;%29=%5Cfrac% 7Bi%7D%7B%5Chbar%7D%5Cleft&space;%28%5Cha t%7BH%7D%5Cpsi%5E%7B%5Cast%7D%5Chat %7BA%7D&space;%5Cpsi-i%5Chbar%5Cpsi%5E%7B%5Cast%7D%5Cfra c%7B%5Cpartial&space;%5Chat%7BA%7D%7D%7B% 5Cpartial&space;t%7D%5Cleft&space;%28&space;%5Cpsi&space;%5 Cright&space;%29-%5Cpsi%5E%7B%5Cast%7D%5Chat%7BA%7D&space; %5Chat%7BH%7D%5Cpsi&space;&space;&space;%5Cright&space;%29


والآن ... نجد أن:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7Bd%7D%7 Bdt%7D%5Cleft&space;%5Clangle&space;%5Chat%7BA% 7D&space;%5Cright&space;%5Crangle=%5Cfrac%7Bi%7 D%7B%5Chbar%7D%5Cleft&space;%28%5Cint%5Ch at%7BH%7D%5Cpsi%5E%7B%5Cast%7D%5Cha t%7BA%7D&space;%5Cpsi%5C;dx-i%5Chbar%5Cint%5Cpsi%5E%7B%5Cast%7D %5Cfrac%7B%5Cpartial&space;%5Chat%7BA%7D% 7D%7B%5Cpartial&space;t%7D%5Cleft&space;%28&space;%5C psi&space;%5Cright&space;%29dx-%5Cint%5Cpsi%5E%7B%5Cast%7D%5Chat%7 BA%7D&space;%5Chat%7BH%7D%5Cpsi&space;&space;&space;dx&space;%5Cr ight&space;%29


يُتبع ...

[رجب مصطفى]

----


وباستعمال رموز ديراك Dirac Notations ... تصبح على الصورة:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7Bd%7D%7 Bdt%7D%5Cleft&space;%5Clangle&space;%5Chat%7BA% 7D&space;%5Cright&space;%5Crangle=%5Cfrac%7Bi%7 D%7B%5Chbar%7D%5Cleft&space;%28%5Cleft&space;%5 Clangle&space;%5Chat%7BH%7D%5Cpsi&space;%5C;&space;%7 C%5C;&space;%5Chat%7BA%7D%5Cpsi&space;%5Cright&space; %5Crangle-i%5Chbar%5Cleft&space;%5Clangle&space;%5Cpsi&space;%5 C;&space;%7C%5C;&space;%5Cfrac%7B%5Cpartial&space;%5C hat%7BA%7D%7D%7B%5Cpartial&space;t%7D&space;%28 %5Cpsi%29%5Cright&space;%5Crangle-%5Cleft&space;%5Clangle&space;%5Cpsi&space;%5C;&space;%7C%5 C;%5Chat%7BA%7D&space;%5Chat%7BH%7D%5Cpsi &space;%5Cright&space;%5Crangle&space;%5Cright&space;% 29


وحيث أن مؤثر الطاقة الكلية، "الهميلتونيان"، هرميتي Hermitian، أي:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cleft&space;%5Clangl e&space;%5Chat%7BH%7D%5Cpsi&space;%5C;&space;%7C%5C;&space; %5Chat%7BA%7D%5Cpsi&space;%5Cright&space;%5Cran gle=%5Cleft&space;%5Clangle&space;%5Cpsi&space;%5C;&space;% 7C%5C;&space;%5Chat%7BH%7D%5Chat%7BA%7D%5 Cpsi&space;%5Cright&space;%5Crangle


فإن:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7Bd%7D%7 Bdt%7D%5Cleft&space;%5Clangle&space;%5Chat%7BA% 7D&space;%5Cright&space;%5Crangle=%5Cfrac%7Bi%7 D%7B%5Chbar%7D%5Cleft&space;%28&space;%5Cleft&space;% 5Clangle&space;%5Cpsi&space;%5C;&space;%7C%5C;&space;%5Chat %7BH%7D%5Chat%7BA%7D%5Cpsi&space;%5Cright &space;%5Crangle-%5Cleft&space;%5Clangle&space;%5Cpsi&space;%5C;&space;%7C%5 C;%5Chat%7BA%7D&space;%5Chat%7BH%7D%5Cpsi &space;%5Cright&space;%5Crangle&space;%5Cright&space;%29+%5 Cleft&space;%5Clangle&space;%5Cpsi&space;%5C;&space;%7C%5C; &space;%5Cfrac%7B%5Cpartial&space;%5Chat%7BA%7D %7D%7B%5Cpartial&space;t%7D&space;%28%5Cpsi%29% 5Cright&space;%5Crangle&space;


ولكن:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cleft&space;[%5Chat%7BH%7D,%5Chat%7BA%7D&space;%5Crigh t&space;]=%5Chat%7BH%7D%5Chat%7BA%7D-%5Chat%7BA%7D%5Chat%7BH%7D


وأيضاً ...


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cleft&space;%5Clangl e&space;%5Cfrac%7B%5Cpartial&space;%5Chat%7BA%7 D%7D%7B%5Cpartial&space;t%7D&space;%5Cright&space;%5C rangle=%5Cleft&space;%5Clangle&space;%5Cpsi%5C; %7C%5C;%5Cfrac%7B%5Cpartial&space;%5Chat% 7BA%7D%7D%7B%5Cpartial&space;t%7D%28%5Cps i%29&space;%5Cright&space;%5Crangle


إذاً ...


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7Bd%7D%7 Bdt%7D%5Cleft&space;%5Clangle&space;%5Chat%7BA% 7D&space;%5Cright&space;%5Crangle=%5Cfrac%7Bi%7 D%7B%5Chbar%7D%5Cleft&space;%5Clangle&space;%5C left&space;[&space;%5Chat%7BH%7D,%5Chat%7BA%7D&space;%5Crig ht&space;]&space;%5Cright&space;%5Crangle+%5Cleft&space;%5Clang le&space;%5Cfrac%7B%5Cpartial&space;%5Chat%7BA% 7D%7D%7B%5Cpartial&space;t%7D&space;%5Cright&space;%5 Crangle


وعندما لا تعتمد الخاصية http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Chat%7BA%7D صراحةً على الزمن، فإن:


http://latex.codecogs.com/gif.latex?%5Clarge&space;%5Cfrac%7Bd%7D%7 Bdt%7D%5Cleft&space;%5Clangle&space;%5Chat%7BA% 7D&space;%5Cright&space;%5Crangle=%5Cfrac%7Bi%7 D%7B%5Chbar%7D%5Cleft&space;%5Clangle&space;%5C left&space;[&space;%5Chat%7BH%7D,%5Chat%7BA%7D&space;%5Crig ht&space;]&space;%5Cright&space;%5Crangle


والله أعلى وأعلم


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