Appendix - Common quantum mechanics and statistical mechanics results

Summary

A review of perturbation theory

Most quantum mechanics problems are not solvable in closed form with analytical techniques. To extend our repertoire beyond just particle-in-a-box, a number of approximation techniques have been developed. A large class of these fall under the heading of “perturbation theory”, in which we consider our system to obey Hamiltonian H that may be written as

H = H⁰ + λH¹ + λ²H² + … , (A.1)

where H⁰ is an exactly solvable Hamiltonian, λ is a small parameter, and the other terms may therefore be taken as small corrections.

These notes are a quick review of how to deal with systems that obey such Hamiltonians. We'll be using Dirac notation and the Schrödinger formalism, in which the states are time-dependent. We begin with time-independent perturbation theory, and will then move on to consider time-dependent problems. For now, we'll only deal with single-particle problems, too.

Time-independent

We're going to use the time-independent Schrödinger equation, and look for the energy eigenvalues and eigenstates of H. First, suppose that our unperturbed problem,

H⁰|ψ⁰〉 = E⁰|ψ⁰〉 (A.2)

may be solved exactly, giving an energy eigenvalue spectrum E⁰_j with corresponding eigenstates |ψ⁰_j〉. Remember, because H⁰ is a Hermitian operator, its eigenvalues are real, and its eigenvectors |ψ⁰_j〉 form a complete set.

The strategy we're going to take is straightforward. We're going to assume that our perturbative corrections to the Hamiltonian, λH¹ + λ²H² + … , lead to corresponding perturbative corrections to the eigenvalues and eigenstates. That is,

E_j = E⁰_j + λE¹_j + λ²E²_j + … ,

|ψj〉 = |ψ⁰_j〉 + λ|ψ¹_j〉 + λ²|ψ²_j〉 + ….