Appendix C - The Wigner–Weisskopf method

Summary

The derivation of the Fermi Golden Rule in Section 9.6.3 is limited to sufficiently short times t ≪ τ₂, and the exponential decay law (9.171) cannot be justified using only the arguments of that section. A method due to Wigner and Weisskopf permits this law to be justified for long times with the help of another approximation scheme. Let us consider the following situation. A state of an isolated system a of energy E_a decays to a continuum of states b of energy E_b. Examples of such a situation are the de-excitation of an excited state of an atom, a molecule, a nucleus, and so on with the emission of a photon, or the decay of an elementary particle. The states of energy E_a and E_b are the eigenstates of a Hamiltonian H⁽⁰⁾:

and a time-independent perturbation W is responsible for the transition a → b; in the case of spontaneous photon emission, W is given by (14.58). The states a and b are not stationary states of the total time-independent Hamiltonian H = H⁽⁰⁾ + W. We can assume that the diagonal matrix elements of W are zero: W_aa = W_bb = 0 and we use |ψ(t)〉 to denote the state vector of the system, the initial state being |ψ(t = 0)〉 = |a〉.