QuantumPhysics.dvi

20 Photon Emission and Absorption

In this chapter, we couple the quantized electro-magnetic field to quantized matter, such as elec-
trons, protons, atoms and molecules. The matter is assumed to be non-relativistic, and thus
described by the usual Hamiltonians where the number of matter particles is fixed. We shall set
up the general problem of photon emission and absorption from matter, calculate the rate for a
single photon, apply this calculation to the case of the 2p state of atomic Hydrogen, and extend
the problem to black body radiation.

20.1 Setting up the general problem of photon emission/absorption

For simplicity, we shall consider here a pure matter HamiltonianHmfor just a single particle of
mass ism, subject to a potentialV,

Hm=

p^2

2 m

+V(x) (20.1)

Concretely, one may think ofHmas describing an electron bound to an atom or to an ion by the
potentialV. We shall denote the Hilbert space of states for the pure matter system byHm, and
denote its states by|ψ〉m. Typically, we shall be interested in the system being initially in the pure
matter state|ψi〉m, which is an eigenstate ofHm, and finally in the pure matter state|ψf〉mplus
photons.

The quantized photons are governed by the Maxwell HamiltonianHEMwhich was studied in
the preceding chapter. It is given by,

HEM=

∫

d^3 x

(

1

2

E^2 +

1

2

B^2

)

(20.2)

The Hilbert space of all photon states will be denoted byHEM. The photon states are labelled
by the momenta and the polarizations of the photons. For example, a state with one-photon of
momentumkand polarizationαis labeled by|k,α〉.

The full Hamiltonian is not just the sum ofHmandHEM, but requires the inclusion of the
couplings between matter and radiation. These interactions are accounted for by including an
interaction HamiltonianHI,

HI=

e

2 m

(A·p+p·A) +

e^2

2 m

A^2 +μB·S (20.3)

For spin 1/2 particles, theB·Sterm naturally arises from the non-relativistic approximation to
the Dirac equation, as we shall establish later. The full hamiltonianH=Hm+HEM+HIthen
takes the form,

H=

1

2 m

(p+eA)^2 +V(x) +μB·S+

∫

d^3 x

(

1

2

E^2 +

1

2

B^2

)

(20.4)

We recognize the first three terms inHas identical to the Hamiltonian for a charged particle in
the presence of an external electro-magnetic field. Thus, the coupling to the quantized fields is