With the foregoing preparation, we are now in a position to
apply the
classical analogy or canonical quantization program
to achieve the second quantization of the electromagnetic
field.[1] As our starting
point and for reference, we, once again, set forth the vacuum or microscopic
Maxwell's equations in the time domain:
 |
[ II-1a ] |
 |
[ II-1b ] |
 |
[ II-1c ] |
 |
[ II-1d ] |
The canonical formulation of classical electrodynamics (Jeans'
Theorem) is most conveniently achieved in terms of the (magnetic)
vector potential in the time domain -- viz.
 |
[ II-2a ] |
 |
[ II-2b ] |
so that
 |
[ II-3a ] |
 |
[ II-3b ] |
In QED (Quantum Electrodynamics) it is convenient
and traditional to make use of the Coulomb
gauge -- i.e. 
-- so that
 |
[ II-4a ] |
 |
[ II-4b ] |
where 
is the so called transversecurrent
density. Since 
is completely determined by the transverse current density in the Coulomb
gauge, electromagnetic problems become in a sense separable -- i.e.
The transversefield
problem:
 |
[ II-5a ] |
The longitudinalfield
problem:
 |
[ II-5b ] |
We turn now explicitly to a treatment of the free
electromagnetic field-- formally the case of 
wherein
 |
[ II-6 ] |
We look for solutions in the form
 |
[ II-7 ] |
Substituting into Equation [ II-6 ] we obtain
 |
[ II-8 ] |
Thus we may apply separation of variables
techniques and the original problem is divided in to two
new and distinct problems -- viz.solutions of the following
set of equations:
 |
[ II-9 ] |
 |
[ II-10 ] |
where the 
's
are separation constants. The spatial
equations allow us to treat the boundary value problem of the cavity or
defining field space in whatever detail that might seem appropriate in
a particular case.[2]
But the bottom lineis that the boundary condition taken together
with the characteristics of the normal modes or eigenfunctions determine
the 
's
which may, in turn, be identified as the eigenfrequencies of the normal
modes.
Thus, we may now write
 |
[ II-11 ] |
and
 |
[ II-12 ] |
The crucial step required in establishing Jeans' Theorem is the expansion
of the instantaneous value of the stored electromagnetic energy in terms
of the cavity modes -- viz.
 |
[ II-13a ] |
which, in light of Equations [ II-11 ] and [ II-12 ], becomes
 |
[ II-13b ] |
To proceed we need the value of the integral 
.
Using a well known vector identity,[3]
we obtain
 |
[ II-14a ] |
By boundary value arguments we may easily show that the first term on
the RHS of this equation vanishes and by using an even more familiar (famous)
vector identity[4] for a divergenceles
field, we obtain
 |
[ II-14b ] |
Therefore, Equation [ II-13b ] becomes
 |
[ II-15 ] |
From Equation [ II-14b ] we may also write
 |
[ II-16a ] |
and, again, by boundary value arguments we may easily show that the
resultant surface imtegral on the left vanishes so that
 |
[ II-16b ] |
Therefore, we may take
 |
[ II-17 ] |
so that Equation [ II-15 ] becomes
 |
[ II-18 ] |
which when compared to Equation [ I-6 ][5]is
effectively the content of Jean's Theorem with
 |
[ II-19 ] |
To accomplish the canonical quantizationprogram,
field variables are expressed as field operators by making the identification
 |
[ II-20 ] |
which leads to the following set of field operators:
 |
[ II-21 ] |
The Plane Wave Expansion of the Electromagnetic
Field Hamiltonian:
To be definite, we may write an explicit plane wave representation for
the field as
 |
[ II-22a ] |
or [6]
 |
[ II-22b ] |
where
 |
[ II-23a ] |
 |
[ II-23b ] |
Of course, 
in all of these expansions. Further the electric field expansion is given
by
 |
[ II-24a ] |
where 
and the magnetic field expansion by
 |
[ II-24b ] |
In light of Equations [ II-18 ] and [ II-21 ], the Hamiltonian of the
radiation field is
 |
[ II-25 ] |
where
 |
[ II-26 ] |
The electromagnetic momentum (Poynting vector divided by c2 )
is given classical by
 |
[ II-27a ] |
and in terms of the second quantization operators it becomes
 |
[ II-27b ] |
Thus, the Fock
or number states are eigenstates of both the energy and the momentum
of the field.
