Subatomic Physics

294 The Electromagnetic Interaction
The classical quantity Hem becomes an operator by translating p and A into
quantum mechanics. The momentum p is straightforward; it becomes the momentum
operator
p −→ −i�∇. (10.55)
This substitution is well known from nonrelativistic quantum mechanics. The corresponding
substitution for A depends on the process under consideration. Two
kinds of emission events occur from the state |α〉. The first takes place in the
presence of an external electromagnetic field, produced, for instance, by photons
incident on the system. A is the field due to these photons, and it gives rise to
stimulated or induced emission of photons. Stimulated photon emission is the basic
physical process involved in lasers. Here we are interested in the second kind of
emission, called spontaneous. Thestate|α〉 can decay even in the absence of an external
electromagnetic field. The expression for A for spontaneous emission cannot
be obtained from nonrelativistic quantum mechanics, because photons are always
relativistic. We circumvent quantum electrodynamics by postulating that A is the
wave function of the created photon. (6) The form of A can be found by considering
the vector potential of a classical electromagnetic plane wave,
A = a0ˆɛ cos(k · x − ωt). (10.56)
Here ˆɛ is the polarization vector and a0 the amplitude. If this wave is contained in
avolumeV , the average energy is given by
or with Eq. (10.38).
W = Vω2 a 2 0
W = V
4π |E|2 ,
4πc2 sin2 (k · x − ωt) = Vω2a2 0
. (10.57)
8πc2 If A is to describe one photon in the volume V , W must be equal to the energy
Eγ = �ω of this photon. This condition fixes the constant a0 as
� �1/2 2 8π�c
a0 =
. (10.58)
ωV
With Eγ = �ω and p γ = �k, the wave function of the photon, Eq. (10.56), is
determined. A is real because classically it is connected to the observable, and
therefore real, fields E and B by Eqs. (10.37) and (10.38). For the application to
emission and absorption it will turn out to be convenient to write Eq. (10.56) into
6 This step can be justified by using quantum electrodynamics. Here we have no choice but to
postulate it without further explanation. See Merzbacher, Chapter 23; Messiah, Section 21.27.