130_notes.dvi

We have taken a step toward quantization of the EM field, at least when we emit or absorb a
photon. With this step, we can correctly compute the EM transitionrates in atoms. Note that we
have postulated that the vacuum has an infinite number of oscillators corresponding to the different
possible modes of EM waves. When we quantize these oscillators, thevacuum has a ground state
energy density in the EM field (equivalent to half a photon of each type). That vacuum EM field
is then responsible for the spontaneous decay of excited states of atoms through the emission of a
photon. We have not yet written the quantum equations that the EM field must satisfy, although
they are closely related to Maxwell’s equations.

29.2 Decay Rates for the Emission of Photons

Ourexpression for the decay rateof an initial stateφiinto some particular final stateφnis

Γi→n=

2 πVni^2

̄h

δ(En−Ei+ ̄hω).

The delta function reminds us that we will have to integrate over final states to get a sensible answer.
Nevertheless, we proceed to include the matrix element of the perturbing potential.

Taking out the harmonic time dependence (to the delta function) asbefore, we have thematrix
element of the perturbing potential.

Vni=〈φn|

e

mc

A~·~p|φi〉= e

mc

[

2 π ̄hc^2

ωV

]^12

〈φn|e−i

~k·~r

ˆǫ·~p|φi〉

We just put these together to get

Γi→n =

2 π

̄h

e^2

m^2 c^2

[

2 π ̄hc^2

ωV

]

|〈φn|e−i

~k·~r

ǫˆ·~p|φi〉|^2 δ(En−Ei+ ̄hω)

Γi→n =

(2π)^2 e^2

m^2 ωV

|〈φn|e−i

~k·~r

ˆǫ·~p|φi〉|^2 δ(En−Ei+ ̄hω)

We must sum (or integrate) over final states. The states are distinguishable so we add the decay
rates, not the amplitudes. We will integrate over photon energies and directions, with the aid of
the delta function. We will sum over photon polarizations. We will sum over the final atomic states
when that is applicable. All of this is quite doable. Our first step is to understand the number of
states of photons as Plank (and even Rayleigh) did to get the Black Body formulas.

29.3 Phase Space: The Density of Final States

We have some experience with calculating the number of states for fermions in a 3D box (See Section
13.1.1). For the box we had boundary conditions that the wavefunction go to zero at the wall of the
box. Now we wish to know how many photon states are in a region ofphase spacecentered on
the wave vector~kwith (small) volume in k-space ofd^3 ~k. (Rememberω=|~k|cfor light.) We will
assume for the sake of calculation that the photons are confined to a cubic volume in position space
ofV=L^3 and imposeperiodic boundary conditionson our fields. (Really we could require the
fields to be zero on the boundaries of the box by choosing a sine wave. The PBC are equivalent to