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NDTIPT - Phys 852 Quantum mechanics II Spring 2008...

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Phys 852, Quantum mechanics II, Spring 2008 Non-Degenerate Time-Independent Perturbation Theory 1/14/2008 Prof. Michael G. Moore, Michigan State University 1 The central problem in time-independent perturbation theory: Let H 0 be the unperturbed (a.k.a. ‘background’, ‘bare’) Hamiltonian whose eigenvalues and eigenvectors are known. Let E (0) n be the n th unperturbed energy eigenvalue, and | n (0) ) be the n th unperturbed energy eigenstate. They satisfy H 0 | n (0) ) = E (0) n | n (0) ) (1) and ( n (0) | n (0) ) = 1 . (2) Let V be a Hermitian operator which ‘perturbs’ the system, such that the full Hamiltonian is H = H 0 + V. (3) The standard approach is to instead solve H = H 0 + λV (4) , and use λ as for book-keeping during the calculation, but set λ = 1 at the end of the calculation. However, if there is a readily identifiable small parameter in the definition of the perturbation operator V , then use it in the place of λ , and do not set it to 1. In these notes we will treat λ as a small parameter, and will not set it equal to unity. The goal is to find the eigenvalues and eigenvectors of the full Hamiltonian (4). Let E n and | n ) be the n th eigenvalue and its corresponding eigenstate. They satisfy H | n ) = E n | n ) (5) and ( n | n ) = 1 . (6) Because λ is a small parameter, it is assumed that accurate results can be obtained by expanding E n and | n ) in powers of λ , and keeping only the leading term(s). Formally expanding the perturbed quantities gives E n = E (0) n + λE (1) n + λE (2) n + . . . , (7) and | n ) = | n (0) ) + λ | n (1) ) + λ 2 | n (2) ) + . . . , (8) where E ( j ) n and | n ( j ) ) are yet-to-be determined expansion coefficients Inserting these expansions into the eigenvalue equation (5) then gives ( H 0 + λV )( | n (0) ) + λ | n (1) ) + λ 2 | n (2) ) + . . . . ) = ( E (0) n + λE (1) n + λ 2 E
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