Quantizing Radiation
Michael Fowler, 5/4/06
Introduction
In analyzing the photoelectric effect in hydrogen, we derived the rate of ionization of a hydrogen atom in a monochromatic electromagnetic wave of given strength, and the result we derived is in good agreement with experiment. Recall that the interaction Hamiltonian was
and we dropped the 
term because it would correspond to the atom giving energy to
the field, and our atom was already in its ground state. However, if we go through the same
calculation for an atom not initially
in the ground state, then indeed an electromagnetic wave of appropriate
frequency will cause a transition rate to a lower energy state, and is the relevant term.
But this is not the whole story. An atom in an excited state
will eventually emit a photon and go to a lower energy state, even if there is zero external field. Our analysis so far does not predict
this—obviously, the interaction written above is only nonzero if 
is nonzero! So what are we missing?
Essentially, the answer is that the electromagnetic field
itself is quantized. Of course, we know
that, it’s made up of photons. Recall
Planck’s successful analysis of radiation in a box: he considered all possible
normal modes for the radiation, and asserted that a mode of energy w could only gain or lose energy in amounts 
. This led to the
correct formula for black body radiation, then Einstein proved that the same
assumption, with the same 
, accounted for the photoelectric effect. We now understand that these modes of
oscillation of radiation are just simple harmonic oscillators, with energy 
, and, just as a mass on a spring oscillator has fluctuations
in the ground state, 
but 
, for these electromagnetic modes 
but 
.
These fluctuations in mean the interaction Hamiltonian is momentarily nonzero, and
therefore can cause a transition.
Therefore, to find the spontaneous transition rate (as it’s
called) for an atom in a zero (classically speaking) electromagnetic field, we
need to express the electromagnetic field in terms of normal modes (we’ll take
a big box), then quantize these modes as quantum simple harmonic oscillators,
introducing raising and lowering operators for each oscillator (these will be
photon creation and annihilation operators) then construct the appropriate
quantum operator expression for to put in the electron-radiation interaction
Hamiltonian.
The bras and kets will now be quantum states of the electron and the radiation field, in contrast to our analysis of the classical field above, where the radiation field didn’t change. (Of course, it did, really, in that it lost one photon, but in the classical limit there are infinitely many photons in each mode, so that wouldn’t register.)
Our treatment follows Sakurai’s Advanced Quantum Mechanics, Chapter 2 (but we stay with ordinary
Gaussian units, so various expressions differ from his by factors of 
).
We use the Coulomb gauge 
, and of course satisfies
.
Taking for convenience periodic boundary conditions in the
big box, we can write (classically) as a
Fourier series at t = 0:
The time-dependence is given by putting in the whole plane
wave: 
, which time dependence can be taken into the coefficient, 
, so
The vector 
is the polarization of
the plane wave. It’s in the same
direction as the electric field.
Actually it varies with 
, because from , it’s perpendicular to . That is, for a given there are two independent polarizations. For along the z-axis, they could be along the x- and y-axes, these would be called linear polarization, and is the
standard approach. But we could also
take the vectors 
. These correspond to circular polarization: equal x-and y-components but with the y-component
90 degrees ahead in phase. You may
recognize the vectors as the eigenvectors
for the rotation operator around the z-axis—the
circularly polarized beam carries angular momentum, 
per photon, pointed
along the direction of motion.
The energy density 
can be expressed as a sum over the individual 
modes.
Writing the electric and magnetic fields in terms of the vector potential,
where
and thereby expressing the total energy 
in terms of the amplitudes 
, then integrating the energy density over the whole large
box the cross terms disappear from the orthogonality of the different modes and
the total energy in the box—the
Hamiltonian—is:
Note that although the Hamiltonian is (of course) time
independent, the coefficients 
here are time dependent, .
But this is formally
identical to a set of simple harmonic oscillators! Recall that for the classical oscillator, 
, the vector 
has time dependence 
, and the oscillator energy is proportional to 
(x, p are the usual
conjugate variables). Clearly, 
here corresponds to 
: same time dependence, same Hamiltonian. Therefore the real and imaginary parts of must also be conjugate variables, which can
therefore be quantized exactly as for the simple harmonic oscillator.
From
we see that the real part of basically gives the contribution of the 
oscillator to 
, and, recalling the time dependence , the imaginary part is proportional to the contribution to 
, that is, to 
. Essentially, then,
the real part of , proportional to the Fourier component of
the vector potential , is what corresponds to displacement x in a 1-D simple harmonic oscillator, and the imaginary part of , the Fourier component of 
, corresponds to the momentum
in the simple harmonic oscillator.
To carry out the quantization, we must express the classical Hamiltonian
in the form
with 
being the imaginary and real parts of the oscillator
amplitude (scaled appropriately)
exactly parallel to the standard treatment of the simple harmonic oscillator:
From the time-dependence , these (classical) variables P, Q are canonical:
.
The Hamiltonian can now be quantized by the standard
procedure. The pairs of canonical
variables P, Q (one pair to each mode ) become operators, the Poisson brackets become commutators,
the scale determined by Planck’s constant:
.
The next step is to express the electron radiation
interaction 
in terms of these
field operators. Since the electromagnetic field is quantized, the interaction
with the electron must be that the electron emits or absorbs quanta
(photons). This is most directly
represented by writing the interaction in terms of creation and annihilation
(raising and lowering) operators:
These satisfy 
(Notice that the annihilation operator 
is nothing but the
operator representation of the classical complex amplitude , with an extra factor to make it dimensionless, 
We discussed this
same equivalence in the lecture on coherent states, which were eigenstates of
the annihilation operator.)
Following the standard simple harmonic oscillator
development, the operator 
has eigenstates with integer eigenvalues, 
, the contribution to the Hamiltonian from the mode is just 
, and 
, 
.
The bottom line is: the classical plane wave expansion of , with wave amplitudes
is replaced on quantization by a parallel operator expansion, the wave amplitude becoming the (scaled)
annihilation operator:
.
Revisiting the Photoelectric Effect, now with a Quantized Field
Recall now that for the photoelectric effect in hydrogen,
following Shankar we wrote the ingoing electromagnetic field 
. The only relevant
component was that going as 
. In this section,
following standard usage (including Shankar) we take an ingoing field 
--an irritating change by a factor of 2, but apparently
unavoidable if we want to follow Shankar’s nonquantized photoelectric effect,
then go on to the quantized case.
Anyway, recall the matrix element to calculate the rate was (with
ingoing wave now 
)
On quantizing the field, from the end of the previous section
(the c at the beginning here being the speed of light).
Now that the electromagnetic field amplitude 
is expressed as an
annihilation operator, appropriate (photon number) bras and kets must be
supplied for it to operate on. The
relevant photon mode is , so labeling the
corresponding photon number states 
the matrix element
that must appear in the Golden Rule is
(We’ve removed the , that just contributes to the 
-function in the Golden Rule.)
Since 
, it is clear that quantizing the incoming electromagnetic
wave amounts to replacing the classical vector potential for this wave
At the photon occupation level 
the (macroscopic)
energy in this single mode 
becomes 
. (Recall the
Hamiltonian for the classical electromagnetic field is in terms of the ’s.)
From , the Golden Rule matrix element
is proportional to 
, so the Golden Rule rate, which includes the square of the
matrix element, will be exactly proportional to . But from , this is proportional to 
, and in fact the
quantum rate of absorption of radiation
is exactly equal to the classical rate over the whole range of field
strengths.
Spontaneous Emission
However, this exact correspondence with the classical result
does not hold for photon
emission! In that case, the atom adds a photon to a mode which already
contains n photons, say, and the
relevant matrix element is , so the equivalent classical vector is 
. This is nonzero even
if is zero—hence spontaneous
emission.
For spontaneous emission, then, the relevant matrix element is
The density of outgoing states for the emitted photon, taking box normalization with periodic boundary conditions as usual, is
so the density of states in energy contribution to the
Golden Rule delta function is 
, and the photon emission rate with polarization 
into a solid angle 
will be:
One slight difference in evaluating the matrix element from our treatment of the photoelectric effect is in the representation of the dipole interaction. Recall that there we gave the equivalent forms
and used the 
representation because the outgoing photoelectron was taken
to be in a plane wave state, an eigenstates of . But for spontaneous
emission, the electron goes from one bound state to another, so the 
form gives a more immediate picture of the interacting dipole
with the external field, and in fact the integration between the states is
generally a little more direct.
So in the matrix element we make the substitution 
, and must then evaluate the atomic matrix element 
. The natural way to
do this is to express the vectors in terms of spherical harmonics, that is, to
write them as spherical vectors,
and similarly for 
The integrals are then
straightforward but tedious (see Shankar page 519, Sakurai (Advanced) page 43 –
but even then, Merzbacher is quoted on the general result.)
An amusing point made by Sakurai is that the total
transition probability for spontaneous emission is 
and this same
expression was obtained using the Correspondence Principle by Heisenberg,
before quantum field theory was invented.
The calculated lifetime of the n = 2 state is 