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lecture 22_perturbation theory

lecture 22_perturbation theory - Chem 120A Spring 2007...

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Chem 120A Perturbation Theory 03/12/07 Spring 2007 Lecture 22 READING: (RS): Chapter 3 (AF): Chapter 6.1-6.7 Introduction to perturbation theory The goal of the perturbation theory is to provide systematic approach for approximating solutions of complex (hard) systems. Suppose that we have an arbitrary Hamiltonian, ˆ H , which we can express as the sum of another Hamiltonian for which we know the exact solutions, ˆ H 0 , and change to the potential term, V , ˆ H = ˆ H 0 + V . (1) The eigenvalues and eigenfunctions of ˆ H 0 are given by ˆ H 0 ψ n = ε n ψ n , where we are using ψ n to indicate a state , vextendsingle vextendsingle ψ n )big , rather than ψ ( x ) which is an amplitude . What we would like to know are the eigenfunctions and eigenvalues of the perturbed Hamiltonian ˆ H . Since V is small, we will seek solutions which are in the form of a perturbative expansion. In order to keep track of the smallness of the perturbation V , it is convenient to introduce a parameter λ , which can take on values between 0 and 1, ˆ H ( λ ) = ˆ H 0 + λ V . (2) Therefore, by adjusting the value of λ , we can adjust the Hamiltonian; when λ = 0 then ˆ H = ˆ H 0 , while when λ = 1, then we have the full system Hamiltonian. The new eigenvalue equation is given by ˆ H ( λ ) Ψ n ( λ ) = E n ( λ ) Ψ n ( λ ) . (3) Since V must be small in order for perturbation theory to be valid, if we plot the eigenenergies of ˆ H ( λ ) vs. λ (Fig. 1), we should see that they are relatively constant, and that they do not cross. This is valid as long as V << ε n . Now, since the V is small, we assume that we can write the eigenvalues and eigenfunction of ˆ H ( λ ) by using a Taylor series expansion in λ : E n ( λ ) = ε n + λε n + λ 2 ε ′′ n + ... Ψ n ( λ ) = ψ n + λψ n + λ 2 ψ ′′ n + ... (4) To first order, the correction to each energy, ε , due to the presence of the perturbation is ε , while ψ n is the first order correction to the eigenfunction. In order to determine what these corrections should be, we will substitute Eqs. 4 into Eq. 3: ( ˆ H 0 + λ V )( ψ n + λψ n + λ 2 ψ ′′ n + ... ) = ( ε n + λε n + λ 2 ε ′′ n + ... )( ψ n + λψ n + λ 2 ψ
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