8.2 First-Order Time-Dependent Perturbation

While Eq. (8.20) is an exact result, we often need to make approximations if we wish to calculate actual probabilities. One way to proceed is to assume a _n( t ? 0) = 1 for the eigenstate n and a _m( t ? 0) = 0 for m ? n and then approximate the value of a _n( t > 0) = 1 by a _n( t = 0) . Such an approximation is called first-order perturbation theory. Typically, this approach is valid when the time-dependent change in potential ( t) is small and thus may be considered a perturbation to the initial system described by the Hamiltonian ₀.

Consider the case in which the system is in an eigenstate n of ₀ at t ? 0, so a _n( t = 0) = 1 and a _m( t = 0) = 0 for m ? n. There is now only one term on the right-hand side of Eq. (8.20) which, for m ? n and t > 0, becomes

since all coefficients a _m( t = 0) = 0 except for a _n( t = 0) = 1. This means that the matrix element W _mn couples n to m and creates the coefficient a _m( t) for times