8.3 Fermi s Golden Rule

We already know from our previous work that a perturbing potential ( t) can bring about transitions between eigenstates of the unperturbed Hamiltonian. If the system is initially prepared in state n and ( t) = 0 for t ? 0, then up to time t = 0 the system remains in state n with energy E _n = ? ? _n. For times t > 0, we allow ( t) ? 0. Hence, when t > 0 it is possible that the system is in a different state m with energy E _m = ? ? _m.

To calculate the transition probability we start with Eq. (8.25)

If the perturbation ( t > 0) ? 0 does not depend on time then the probability of scattering out of state n into state m as a function of time is

where ? ? _mn = E _m ? E _n and W _mn = m n . Since

Eq. (8.45) may be written

We now take the long time limit so that t ? ? and make use of the relationship

so that

Making use of the fact

if we set x = ? _mn /2 and a = 2 ?, then ?( ? _mn /2) = 2 ? ?(