obtained by simply promoting the classicalAfield to a quantum operator. The Hilbert spaceHof
the full system is the tensor product of the Hilbert spaces ofthe matter and free radiation parts,
and given byH=Hm⊗HEM.
20.2 Single Photon Emission/Absorption
Having developed the general set-up of the matter-radiation system, we shall now apply it to
the emission/absorption of a single photon. The initial state is assumed to correspond to a pure
matter state|ψi〉mofHm, and to have no photons. We shall denote this total state by|ψi;∅〉=
|ψi〉m⊗|∅〉EM. We shall assume that|ψi〉mis an eigenstate ofHm. This condition often constitutes
a good approximation to the situation in a real physical problem, in view of the fact that the electro-
magnetic couplingα=e^2 / ̄hcis small. The final state corresponds to a pure matter state|ψf〉m,
but now has an extra photon. We shall denote this state by|ψf;k,α〉=|ψf〉m⊗|k,α〉EM, and
assume that also|ψf〉mis an eigenstate ofHm. Since the coupling is weak, we shall use first order
perturbation theory to evaluate the transition rate. To do so, the full Hamiltonian is separated
into a “free part”H 0 =Hm+HEMand an interacting partHI. The initial states are eigenstates
ofH 0 ,
H 0 |ψi;∅〉 = Ei(0)|ψi;∅〉
H 0 |ψf;k,α〉 = (Ef(0)+ωk)|ψf;k,α〉 (20.5)whereE(0)i andEf(0)are the energies of|ψi〉mand|ψf〉mrespectively, andωk=|k|.
First order perturbation theory allows us to compute the transition rate Γi→ffrom the initial
state|i〉=|ψi;∅〉to the final state|f〉=|ψf;k,α〉, under the time evolution governed by the full
HamiltonianH=H 0 +H 1. The rate is given by Fermi’s golden rule formula,
Γi→f= 2π∣∣
∣〈f|H 1 |ı〉∣∣
∣2
δ(Ef−Ei) (20.6)whereEf andEiare the total energies of the final and initial states respectively, and given by
Ei=Ei(0)andEf =Ef(0)+ωk. The rate Γi→f is often referred to as theexclusiverate because
the final state is specified to be a photon with specific momentumkand polarizationα. It is often
useful to consider theincludiveortotal rate, obtained by summing over the contributions to all
possible final states, obtained here by summing over all photon momenta and polarizations. The
total rate is given by
Γ =∑fΓi→f (20.7)To evaluate the rate in practice, we have to calculate the matrix element ofH 1 between the state
without photon and the state with one photon. As a result, theA^2 term inH 1 does not contribute,
since it has vanishing matrix elements. The term involving the electron magnetic moment is
typically much smaller than the contribution from the orbital term, and will be neglected here. (Of