This can be used to easily compute decay rates for Hydrogen, for example the 2p decay rate.
Γ 2 p→ 1 s=4 αωin^3
9 c^2∣
∣
∣
∣
∣
4
√
6
(
2
3
) 5
a 0∣
∣
∣
∣
∣
2The total decay rate is related to the energy width of an excited state, as might be expected from
the uncertainty principle. The Full Width at Half Maximum (FWHM) of the energy distribution of
a state is ̄hΓtot. The distribution in frequency follows a Breit-Wigner distribution.
Ii(ω) =|φi(ω)|^2 =1
(ω−ω 0 )^2 +Γ
2
4In addition to the inherent energy width of a state, other effects can influence measured widths,
including collision broadening, Doppler broadening, and atomic recoil.
The quantum theory of EM radiation can be used to understand many phenomena, including photon
angular distributions, photon polarization, LASERs, the M ̈ossbauer effect, the photoelectric effect,
the scattering of light, and x-ray absorption.
1.37 Classical Field Theory
A review of classical field theory (See section 31) is useful to ground our development of relativistic
quantum field theories for photons and electrons. We will work with 4-vectors like the coordinate
vector below
(x 1 ,x 2 ,x 3 ,x 4 ) = (x,y,z,ict)
using theito get a−in the time term in a dot product (instead of using a metric tensor).
A Lorentz scalar Lagrangian density will be derived for each field theory we construct. From the
Lagrangian we can derive a field equation called the Euler-Lagrange equation.
∂
∂xμ(
∂L
∂(∂φ/∂xμ))
−
∂L
∂φ= 0
The Lagrangian for a massive scalar fieldφcan be deduced from the requirement that it be a scalar
L=−
1
2
(
∂φ
∂xν∂φ
∂xν+μ^2 φ^2)
+φρwhere the last term is the interaction with a source. The Euler-Lagrange equation gives
∂
∂xμ∂
∂xμ
φ−μ^2 φ=ρwhich is the known as theKlein-Gordon equationwith a source and is a reasonable relativistic
equation for a scalar field.
Using Fourier transforms, the field from a point source can be computed.
φ(~x) =−Ge−μr
4 πr