The Interaction of Radiation and Matter: Quantum Theory (cont.)

II. Canonical Quantization of Electrodynamics (pdf)
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.

[1] In common usage, the process of treating the cordinates  and  as quantized variables is called first quantization. Second quantizationis the process of quantizing fields -- say,  -- which have an infinite number of dequees of freedom.

[2] The  's  do not form a true complete set of solutions since no longitudinal vector field can be expanded in terms of such divergentlessfunctions.

[3] Namely, that

[4] Namely, that

[6] This expansion is often written as

where
and