Find the rate of transitions from initial state |i〉 with N_p phonons in state p(vector) and N_q in state q(vector), and none in any other states, to the final many-body state |f〉 with N_p − 1 phonons in state p(vector), and N_q − 1 in state q(vector), and one phonon in state p(vector)′, for the interaction term

where

This type of term for inelastic phonon scattering will be derived in Section 5.4. Note that the total number of phonons is not conserved.

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Vint=Σ(Aaaa₁ + A²a²a²a₁), ‚a-a-a- kk k k k k,k,k" (4.7.22)