Introduction

Suppose that the Hamiltonian of the system under consideration can be written

$$H = H_0 + H_1(t),$$ (739)

where $H_0$ does not contain time explicitly, and $H_1$ is a small time-dependent perturbation. It is assumed that we are able to calculate the eigenkets of the unperturbed Hamiltonian:

$$H_0 \,\vert n\rangle = E_n \,\vert n\rangle.$$ (740)

We know that if the system is in one of the eigenstates of $H_0$ then, in the absence of the external perturbation, it remains in this state for ever. 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.