20.3 Application to the decay rate of 2p state of atomic Hydrogen
The simplest application of the above general formulation of single photon decay is to the decay of
the 2p state of atomic Hydrogen to its 1s ground state,|ψi〉=| 2 p〉, and|ψf〉=| 1 s〉. We have,
|〈 1 s|z| 2 p〉|^2 =(
8
9) 5
1
m^2 α^2
E 2 (0)p −E(0) 1 s =3
8
mc^2 α^2 (20.22)Combining these partial results, we find,
Γ =mc^2 α^5 τ=
̄h
Γ= 1. 6 × 10 −^9 sec (20.23)whereτstands for the life-time.
This is one instance where the rules of the Wigner-Eckart theorem come in handy. The position
operatorxis a vector orj= 1 operator, and thus we know that the initial and final total angular
momenta, respectivelyjiandjf, must obey,
∣∣
∣ji−jf
∣∣
∣≤ 1 (20.24)In addition, there is a selection rule on the conservation ofmagnetic quantum number,
mi=mf± 1 (20.25)What happens for initial and final states for which these conditions are not satisfied? Is the state
|ψi〉then stable?
The answer to these questions can be gathered by realizing that in the above calculation, we
have made various approximations to the real situation. We considered only single photon decay,
and neglected the electron magnetic moment coupling.
When two or more photons are exchanged, the difference in totalangular momentum is pretty
much arbitrary. Every additional photon produced, however, will require an extra factor ofein the
amplitude and thus an extra factor ofαin the rate. Sinceα∼ 1 /137, the production of multiple
photons will be suppressed. Also, for multiple photons, less phase space becomes available, and in
general a suppression will result from phase space considerations as well. In summary, when the
above selection rules for single photon decay are not satisfied, the rate will be suppressed, and the
life-time of the 2p state will be prolonged.
20.4 Absorption and emission of photons in a cavity
By acavity, we mean an enclosure whose walls are built of solid material, such as a metal or a
porcelain. The atoms and molecules of the cavity wall vibrate (increasingly so when the tempera-
ture increases), and emit and absorb photons. At equilibrium, a balance is achieved where photons
are being absorbed and emitted in a steady state process. Thespectrum of photons (including the