QuantumPhysics.dvi

and we find the following formulas for respectively absorption and emission,

〈nα(k)− 1 ,α,k|A(t,x)|nα(k),α,k〉 = (nα(k))

1

(^2) ε∗α(k)eik·x
〈nα(k) + 1,α,k|A(t,x)|nα(k),α,k〉 = (nα(k) + 1)
(^12)
εα(k)e−ik·x (20.32)
The extra factor of (nα(k))^1 /^2 is the key difference with the calculation of the single-photon emission
rate. The differential rate for the absorption of a single photon is then,
dΓabs=
8 π^2
L^3
nα(k)
α
2 ωkm^2
∣∣
∣∣〈ψf|εα(k)·peik·x|ψi〉
∣∣
∣∣
2
δ(Ef−Ei−ωk)dωk (20.33)
while the rate for the emission of a photon is given by,
dΓemi=
8 π^2
L^3
(nα(k) + 1)
α
2 ωkm^2
∣∣
∣∣〈ψf|ε∗α(k)·pe−ik·x|ψi〉
∣∣
∣∣
2
δ(Ef−Ei+ωk)dωk (20.34)

20.5 Black-body radiation

Black body radiation assumes thermodynamic equilibrium between the emitted and absorbed pho-
tons and the cavity wall. The process responsible for this thermalization may be depicted as
follows,

ψA↔ψB+γ (20.35)

The absorption and emissions rates, calculated previously, may be applied to this process alterna-
tively with the initial state|ψi〉being|ψA〉(for emission), and|ψB〉for absorption, and the final
state|ψf〉being|ψA〉for absorption, and|ψB〉for emission.

The corresponding emission and absorption rates for the above process are then given by

wemi(α,k) = (nα(k) + 1)

∣∣

∣∣〈ψB|ε∗α(k)·pe−ik·x|ψA〉

∣∣

∣∣

2

wabs(α,k) = nα(k)

∣∣

∣∣〈ψA|εα(k)·peik·x|ψB〉

∣∣

∣∣

2

(20.36)

The matrix elements are related by complex conjugation,

〈ψB|ε∗α(k)·pe−ik·x|ψA〉=〈ψA|εα(k)·peik·x|ψB〉∗ (20.37)

This relation requires commutingpwith the exponential; since the commutator [p,eik·x] = ̄hk,
andεα(k)·k= 0, this commutator does not contribute. Since the rates involve the absolute value
square of these quantities, we find that

wemi(α,k)

nα(k) + 1

=

wabs(α,k)

nα(k)

(20.38)