Introduction

Suppose that the Hamiltonian of a given quantum mechanical system can be written

$$H = H_0 + H_1(t),$$ (8.1)

where the unperturbed Hamiltonian, $H_0$ , does not depend on time explicitly, and $H_1(t)$ is a small time-dependent perturbation. In the following, it is assumed that we are able to calculate the eigenkets and eigenvalues of the unperturbed Hamiltonian exactly. These are denoted

$$H_0\, \vert n\rangle = E_n\, \vert n\rangle,$$ (8.2)

where $n$ is an integer. We know that if the system is initially in one of the eigenstates of $H_0$ then, in the absence of the external perturbation, it remains in that state for ever. (See Section 3.5.) However, the presence of a small time-dependent perturbation can, in principle, give rise to a finite probability that a system initially in some eigenstate $\vert i\rangle$ of the unperturbed Hamiltonian is found in some other eigenstate at a subsequent time (because $\vert i\rangle$ is no longer an exact eigenstate of the total Hamiltonian). In other words, a time-dependent perturbation allows the system to make transitions between its unperturbed energy eigenstates. Let us investigate such transitions.