Subatomic Physics

298 The Electromagnetic Interaction
ˆɛ must be in the xy plane. With the angles θ and ϕ as defined in Fig. 10.6, the components
of 〈β|x|α〉 and ˆɛ are 〈β|x|α〉 = |〈β|x|α〉|(sin θ, 0, cos θ), ˆɛ =(cosϕ, sin ϕ, 0).
Performing the scalar product in Eq. (10.74) then gives
dwβα = e2
2π� 4 c 3 E3 γ |〈β|x|α〉|2 sin 2 θ cos 2 ϕdΩ. (10.75)
If the polarization of the emitted photon is not observed, dwβα must be integrated
over the angle ϕ and summed over the two polarization states. The sum introduces
afactor2;with
dΩ =sinθdθdϕ and
� 2π
the transition rate for an unpolarized photon becomes
0
dϕ cos 2 ϕ = π,
dwβα = e2
� 4 c 3 E3 γ|〈β|x|α〉| 2 sin 3 θdθ. (10.76)
The total transition rate wβα is obtained by integration over dθ,
wβα =
� π
The lifetime (mean life) is the reciprocal of wβα.
0
dwβα = 4 e
3
2
�4c3 E3 γ |〈β|x|α〉|2 . (10.77)
The physical content of the expression (10.77) for the total transition rate becomes
more transparent if appropriate units are introduced. If the decaying system
or particle has a mass m, then the characteristic length associated with it is the
Compton wavelength, λc = �/mc, andE0 = mc 2 is the characteristic energy. The
time that it takes light to move the distance λc is given by t0 = �/mc 2 ,andthe
inverse of this time, w0 =1/t0 = mc 2 /�, is the characteristic transition rate. With
λc,E0 = mc 2 ,andw0, the transition rate is rewritten as
wβα
w0
= 4
3
� e 2
�c
��
Eγ
mc2 �3 2 |〈β|x|α〉|
λ 2 . (10.78)
c
The transition rate, expressed in terms of the “natural” rate w0, becomes a product
of three dimensionless factors, each of which has a clear physical interpretation.
The last term, |〈β|x|α〉| 2 /λ 2 c , contains the information about the structure of the
decaying system. If the wave functions Φα and Φβ are known, the electric dipole
matrix element 〈β|x|α〉 can be computed. Even without calculation, however, some
properties can be deduced. For instance, the states |α〉 and |β〉 must have opposite
parities; otherwise 〈β|x|α〉 vanishes, and no electric dipole radiation can be emitted.