lecture23_umn

lecture23_umn - Lecture 23 November 4 2009...

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Unformatted text preview: Lecture 23 November 4, 2009 Rayleigh-Schrödinger perturbation theory Perturbation Theory Exact eigenfunctions, eigenvalues of a simplified operator: possible to estimate eigenfunctions, eigenvalues of a more complete operator. Rayleigh-Schrödinger perturbation theory Some operator A that we can write as (23-1) A (0) is an operator for which we can find eigenfunctions V is a perturbing operator and λ is a dimensionless parameter that, as it varies from 0 to 1, maps A (0) into A . A = A (0) + " V Taylor Series Expansion (23-2) and (23-3) is the eigenvalue for , appropriate normalized ground-state eigenfunction for A (0) . Ease of notation, eqs. 23-2 and 23-3 (23-4) and (23-5) superscripts ( n ) n th-order corrections to the zeroth order term, defined by comparison to eqs. 23-2 and 23-3. " = " (0) + # $" (0) $# # = + 1 2! # 2 $ 2 " (0) $# 2 # = + 1 3! # 3 $ 3 " (0) $# 3 # = + L a = a (0) + " # a (0) #" " = + 1 2!...
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lecture23_umn - Lecture 23 November 4 2009...

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