Time-dependent perturbation theory

We treat the hamiltonian as a zero-order part that does not depend on time, H₀ and a perturbation H' that does. The Schrödinger equation then takes the form

We assume that the zero-order eigenfunctions and eigenvalues are known. Defining y_n(0) º |nñ as eigenfunctions of H, H|nñ = E|nñ, we can use these zero-order solutions as a basis for expanding the perturbed wave function.

The coefficients c_n(t) and phase factor e^-iEt/h carry the time dependence. The coefficients give the time-dependent probability amplitudes for the zero order states n. We substitute this first-order expression into the Schrödinger equation to obtain

The sum over n runs over an infinite number of eigenstates. If we pick one of these states (call it m) and multiply both sides by the complex conjugate of the wave function of state m and then integrate over all space we find

The eigenfunctions are orthonormal án|mñ = d_nm so out of the infinite sum on the right-hand side only the term in n = m survives.

Since

we have

Note that there are terms on each side that cancel and thus we can write a set of coupled linear equations for the coefficients

where some standard definitions have been used:

The solution thus far is exact. However, because the coefficients comprise a set of linear coupled equations the solution is still not practical. We can obtain a perturbation theory solution by assuming that all of the coefficients on the right hand side are equal to their values at t = 0, c_n(t) » c_n(0) = d_ni. This eliminates all of the terms in the summation on the right hand side except one. We assume that the wave function starts out in an initial state i and thus at time zero all of the population is in I (c_i(0) = 1). We wish to know the time dependence of the coefficient for a final state f. We can integrate directly to find this.

The coefficient gives a probability amplitude. The probability is the square of the wave function (Y*Y = Y² if Y is real). Thus, the probability for observing population in state f at time t is