量子力學(xué)-斯坦福大學(xué)-chapter09--資料下載頁
2025-08-05 17:52本頁面
【正文】 E c?????? is a system of m of equations for the cj’s. ? ?1 1 1 1 2 2 1 0mmH E c H c H c? ? ? ?? ? ? ? ?? ?2 1 1 2 2 2 2 0mmH c H E c H c? ? ? ?? ? ? ? ?? ?1 1 2 2 0m m m m mH c H c H E c? ? ? ?? ? ? ? ?One equation for each index i of ci. 12 0mc c c? ? ? ?Besides trivial solution of only get solution if the determinant of the coefficients vanish. ? ?? ?? ?11 12 122 2120mmm m m mH E H HH E HH H H E? ? ? ??? ? ???? ? ? ??00jk j kHH?????We know the Have mth degree equation for the . Es?Copyright – Michael D. Fayer, 2022 Solve mth degree equation – get the . Now have the corrections to energies. ? ?1 1 1 1 2 2 1 0mmH E c H c H c? ? ? ?? ? ? ? ?? ?2 1 1 2 2 2 2 0mmH c H E c H c? ? ? ?? ? ? ? ?? ?1 1 2 2 0m m m m mH c H c H E c? ? ? ?? ? ? ? ?iEs?iE?To find the correct zeroth order eigenvectors, one for each , substitute (one at a time) into system of equations. iE?Get system of equations for the coefficients, cj’s. 1 1 2 2 1* * *, mmc c c c c c? ? ? ?There are only m – 1 conditions because can multiply everything by constant. Use normalization for mth condition. Now we have the correct zeroth order functions. ijH?Know the . Copyright – Michael D. Fayer, 2022 1 , 2 , mE E E? ? ?01 1iiE E E i m?? ? ? ?The solutions to the mth degree equation (expanding determinant) are Therefore, to first order, the energies of the perturbed initially degenerate states are Have m different (unless some still degenerate). iEs?01W it ha s 0iEE???Copyright – Michael D. Fayer, 2022 ? ?00 11mk k j k jjE E A c H??? ? ? ?Correction to wavefunctions Again using equation found substituting the expansions into the first order equation ? ?0 0 0 0 0111mmk k k j j j k j kk j k jE E A E c c H? ? ???????? ? ?????? ? ? ?0i i k m? ??Left multiply by Orthogonality makes other terms zero. Normalization gives 1 for nonzero brackets. gives 1 gives 0 ? ?1001mj kjjkkcHA k mEE??????Therefore Normalization gives Aj = 0 for j ? m. Already have part of wavefunction for j ? m Copyright – Michael D. Fayer, 2022 First order degenerate perturbation theory results ? ?100001mj kjji i kkm kcHEE? ? ? ????? ? ????01iiE E E? ?? ? ?Correct zeroth order function. Coefficients ck determined from system of equations. Correction to zeroth order function. Copyright – Michael D. Fayer, 2022
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