Subatomic Physics

10.4. Photon Emission 295
the form
� �1/2 � � �
2 2 2π� c i(pγ · x − Eγt)
A(one photon) =
ˆɛ exp
EγV
�
� ��
−i(pγ · x − Eγt)
+exp
. (10.59)
�
Here, A is no longer a classical vector potential, but it is postulated to be the wave
function of the emitted photon. A is a vector, as is appropriate for photons which
are spin-1 particles (Section 5.5). The next step is the construction of the matrix
element of Hem,
�
〈β|Hem|α〉 ≡
= e
mc
d 3 xψ ∗ βHemψα
�
d 3 xψ ∗ βpψα · A = −i e�
mc
�
d 3 xψ ∗ β∇ψα · A. (10.60)
To evaluate 〈β|Hem|α〉, we make approximations. The first is the electric dipole
approximation. The momentum part of the exponent in A can be expanded,
� �
±ipγ·x
exp
�
=1± i p γ · x
�
+ ··· . (10.61)
The exponential can be replaced by unity if p γ · x ≪ �. To obtain an approximate
idea of what this condition implies, we assume that x has roughly the size of the
system that emits the photon, and we denote this dimension by R. The condition
imposed on the gamma-ray energy then is
Eγ = pγc ≪ �c
R
197 MeV-fm
� . (10.62)
R(in fm)
The second approximation applies to the decaying system. We assume it to be
spinless and so heavy that it is at rest before and after the emission of the photon.
The wave functions ψα and ψβ canthenbewrittenas
� �
−iEαt
ψα(x,t)=Φα(x)exp
�
� �
−iEβt
ψβ(x,t)=Φβ(x)exp ,
�
(10.63)
where Φα(x) andΦβ(x) describe the spatial extension of the system before and
after the photon emission (Chapter 6). Eα and Eβ are the rest energies of the
initial and final states. Energy conservation demands that
Eα = Eβ + Eγ. (10.64)