Advanced Undergraduate Quantum Mechanics pp 429-463 | Cite as

# Perturbation Theory for Stationary States: Stark Effect and Polarizability of Atoms

## Abstract

Only few models in quantum mechanics allow for an exact analytical solution. Most of the problems, which are relevant to the real-world situations and are important for understanding the fundamental nature of things or for applications, can only be solved using one or another type of approximation. In this chapter I will introduce a method designed for finding approximate solutions for eigenvalues and corresponding eigenvectors of a time-independent Hamiltonian with a discrete spectrum. This method works for Hamiltonians that can be written as a sum of two parts: the main or unperturbed Hamiltonian \(\hat {H}_{0}\), whose eigenvalues, \(E_{s}^{(0)}\), and eigenvectors, \(\left |s\right \rangle \), are presumed to be known, and a perturbation \(\epsilon \hat {V}\). Parameter *𝜖* that I pulled out of \(\hat {V}\) has a formal meaning of the strength of the perturbation, but this can be understood literally only in the sense that the perturbation vanishes when *𝜖* = 0. The actual parameter determining the strength of the perturbation emerges only *post factum*, after the problem is solved. I will mainly use *𝜖* as a technical bookkeeping device (you will know what it means when you see it) and set it equal to unity at the end. The index *s* appearing in the notation for the eigenvalues and the eigenvectors can be a composite index, consisting of several subindexes. For instance, if \(\hat {H}_{0}\) is the Hamiltonian of a hydrogen-like atom, then *s* contains principal, orbital, and magnetic numbers *n*, *l*, *m*. It is also presumed that the perturbation \(\hat {V}\) can be considered small in some yet undefined sense so that the eigenvalues and eigenvectors of the total Hamiltonian