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Unformatted text preview: X. Perturbation Theory In perturbation theory, one deals with a Hamiltonian that is composed of two pieces: a reference part ( ) ˆ H that is typically exactly solvable and a perturbation ( ) 1 ˆ H that is assumed to be “small”. In practice, this usually arises because we can experimentally control the importance of 1 ˆ H ; for example, if 1 ˆ H represents the interaction with an external magnetic field, we can control the strength of this interaction by varying the magnitude of the field. In this general situation, it is useful to consider the general Hamiltonian: ( ) 1 ˆ ˆ ˆ H H H λ λ + = Here, λ is our “control” parameter – it allows us to isolate the influence of 1 ˆ H on the eigenvalues and eigenstates of ( ) λ H ˆ . At the end of the calculation, the “physical” Hamiltonian will always correspond to 1 = λ , but at the intermediate stages λ allows us to collect terms in a meaningful way. Now, given an arbitrary Hamiltonian, how are we to choose the appropriate reference Hamiltonian? It is clear that for a given H ˆ we can choose any reference we like by writing: ( ) ( ) ˆ ˆ ˆ ˆ H H H H- + = λ λ Hence, if we define 1 ˆ ˆ ˆ H H H- = the full Hamiltonian takes the desired reference+perturbation form for any choice of ˆ H ! In practice this is complicated by the fact that different choices of the reference ˆ H will give different perturbation expansions – a good choice will give accurate answers, but a bad choice will give poor results. Thus, in many situations, the accurate use of perturbation theory essentially reduces to the art of choosing a good reference ˆ H . However, in this course we will assume that we know the exact eigenvalues and eigenstates of ˆ H : ( ) ( ) ( ) ˆ n n n E H ψ ψ = . This severely limits our choices of ˆ H , since at present we only know two exactly solvable Hamiltonians (the Harmonic oscillator and the piecewise constant potential). Hence, every problem we treat will look like (Harmonic oscillator + other terms) or (step potential + other terms). It should be stressed that this is not a necessary assumption to apply perturbation theory; one can also formulate perturbative expansions based on approximately solvable reference Hamiltonians, but we will not treat this case in this course. Given that we know the eigenstates and eigenvalues of ˆ H , we now seek to understand how 1 ˆ H influences these eigenvalues and eigenstates. Thus, we are interested in the solutions of the equation: ( ) ( ) ( ) ( ) λ ψ λ λ ψ λ n n n E H = ˆ Now, recall that our physical picture is that 1 ˆ H has a “small” influence on ˆ H . This can be enforced by examining the behavior of the eigensystem for small λ . To this end, we assume that we can expand the eigenstates and eigenvalues in a Taylor series in λ : ( ) ( ) ( ) ( ) ......
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
- Spring '04