130_notes.dvi

29 Radiation in Atoms

Now we will go all the way back toPlankwho proposed that the emission of radiation be in quanta
withE= ̄hωto solve the problem of Black Body Radiation. So far, in our treatment of atoms, we
have not included the possibility toemit or absorb real photonsnor have we worried about the
fact that Electric and Magnetic fields are made up of virtual photons. This is really the realm of
Quantum Electrodynamics, but we do have the tools to understandwhat happens as we quantize
the EM field.

We now have the solution of the Harmonic Oscillator problem using operator methods. Notice that
theemission of a quantum of radiationwith energy of ̄hωislike the raising of a Harmonic
Oscillator state. Similarly the absorption of a quantum of radiation is like the lowering ofa HO
state. Plank was already integrating over an infinite number of photon (like HO) states, the same
integral we would do if we had an infinite number of Harmonic Oscillator states. Plank was also
correctly counting this infinite number of states to get the correct Black Body formula. He did it
by considering a cavity with some volume, setting the boundary conditions, then letting the volume
go to infinity.

This material is covered inGasiorowicz Chapter 22,inCohen-Tannoudji et al. Chapter
XIII,and briefly in Griffiths Chapter 9.

29.1 The Photon Field in the Quantum Hamiltonian

The Hamiltonian for a charged particle in an ElectroMagnetic field (SeeSection 20.1) is given by

H=

1

2 m

(

~p+

e

c

A~

) 2

+V(r).

Letsassume that there is some ElectroMagnetic field around the atom. The field is not
extremely strong so that theA^2 term can be neglected (for our purposes) and we will work in the
Coulomb gaugefor which~p·A~=h ̄i∇·~ A~= 0. The Hamiltonian then becomes

H≈

p^2

2 m

+

e

mc

A~·~p+V(r).

Now we havea potentially time dependent perturbationthat may drive transitions between
the atomic states.
V=

e

mc

A~·~p

Lets also assume that the field hassome frequencyωand corresponding wave vector~k. (In fact,
and arbitrary field would be a linear combination of many frequencies,directions of propagation,
and polarizations.)
A~(~r,t)≡ 2 A~ 0 cos(~k·~r−ωt)

whereA~ 0 is a real vector and we have again introduced the factor of 2 for convenience of splitting
the cosine into two exponentials.

We need to quantize the EM field into photons satisfying Plank’s original hypothesis,E= ̄hω. Lets
start by writingAin terms of the number of photons in the field(at frequencyωand wave