130_notes.dvi

In a gauge in which∇·~ A~= 0, the perturbation is

V=

e

mc

A~·~p+ e

2

2 mc^2

A^2.

For most atomic decays, theA^2 term can be neglected since it is much smaller than theA~·~pterm.
Both the decay of excited atomic states with the emission of radiation and the excitation of atoms
with the absorption of radiation can be calculated.

An arbitrary EM field can be Fourier analyzed to give a sum of components of definite frequency.
Consider the vector potential for one such component,A~(~r,t)≡ 2 A~ 0 cos(~k·~r−ωt). The energy in

the field isEnergy= ω

2

2 πc^2 V|A^0 |

(^2). If the field is quantized (as we will later show) with photons of
energyE= ̄hω, we may write field strengthin terms of the number of photonsN.
A~(~r,t) =

[

2 π ̄hc^2 N

ωV

]^12

ˆǫ2 cos(~k·~r−ωt)

A~(~r,t) =

[

2 π ̄hc^2 N

ωV

]^12

ˆǫ

(

ei(

~k·~r−ωt)

+e−i(

~k·~r−ωt))

The direction of the field is given by the unit polarization vector ˆǫ. The cosine term has been
split into positive and negative exponentials. In time dependent perturbation theory, the positive
exponential corresponds to the absorption of a photon and excitation of the atom and the negative
exponential corresponds to the emission of a photon and decay ofthe atom to a lower energy state.

Think of the EM field as a harmonic oscillator at each frequency, the negative exponential corre-
sponds to a raising operator for the field and the positive exponential to a lowering operator. In
analogy to the quantum 1D harmonic oscillator we replace

√

Nby

√

N+ 1 in the raising operator
case.

A~(~r,t) =

[

2 π ̄hc^2

ωV

]^12

ˆǫ

(√

Nei(

~k·~r−ωt)

+

√

N+ 1e−i(

~k·~r−ωt))

With this change, which will later be justified with the quantization of the field, there is apertur-
bation even with no applied field(N= 0)

VN=0=VN=0eiωt=

e

mc

A~·~p= e

mc

[

2 π ̄hc^2

ωV

]^12

e−i(

~k·~r−ωt)

ˆǫ·~p

which can cause decays of atomic states.

Plugging thisN= 0 field into the first order time dependent perturbation equations, the decay rate
for an atomic state can be computed.

Γi→n =

(2π)^2 e^2

m^2 ωV

|〈φn|e−i~k·~rˆǫ·~p|φi〉|^2 δ(En−Ei+ ̄hω)

The absolute square of the time integral from perturbation theory yields the delta function of energy
conservation.

To get the total decay rate, we must sum over the allowed final states. We can assume that the
atom remains at rest as a very good approximation, but, the final photon states must be carefully