11: Time-Independent Perturbation Theory

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Consider the following very commonly occurring problem. The Hamiltonian of a quantum mechanical system is written \[H = H_0 + H_1.\] Here, \(H_0\) is a simple Hamiltonian whose eigenvalues and eigenstates are known exactly. \(H_1\) introduces some interesting additional physics into the problem, but is sufficiently complicated that when we add it to \(H_0\) we can no longer find the exact energy eigenvalues and eigenstates. However, \(H_1\) can, in some sense (which we shall specify more precisely later on), be regarded as small compared to \(H_0\). Can we find approximate eigenvalues and eigenstates of the modified Hamiltonian, \(H_0+H_1\), by performing some sort of perturbation expansion about the eigenvalues and eigenstates of the original Hamiltonian, \(H_0\)? Let us investigate.

Incidentally, in this chapter, we shall only discuss so-called time-independent perturbation theory, in which the modification to the Hamiltonian, \(H_1\), has no explicit dependence on time. It is also assumed that the unperturbed Hamiltonian, \(H_0\), is time independent.