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

With the foregoing preparation, we are now in a position to apply theclassical analogyorcanonical quantizationprogram to achieve theof 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:second quantization

[ II-1a ]

[ II-1b ]

[ II-1c ]

[ II-1d ] The canonical formulation of classical electrodynamics (

) is most conveniently achieved in terms of the (magnetic) vector potential in the time domain --Jeans' Theoremviz.

[ II-2a ]

[ II-2b ] so that

[ II-3a ]

[ II-3b ] In QED (Quantum Electrodynamics) it is

to make use of theconvenient and traditionalCoulomb gauge --i.e.-- so that

[ II-4a ]

[ II-4b ] where is the so called

current density. Since is completely determined by the transverse current density in the Coulomb gauge, electromagnetic problems become in a sense separable --transversei.e.

Thetransversefield problem:

[ II-5a ] Thelongitudinalfield problem:

[ II-5b ] We turn now explicitly to a treatment of the

-- formally the case of whereinfree electromagnetic field

[ 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 variablestechniques and the original problem is divided in totwo new and distinctproblems --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 thebottom 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 Theoremwith

[ II-19 ]

To accomplish the, field variables are expressed as field operators by making the identificationcanonical quantizationprogram

[ 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 c

^{2 }) 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
quantization*is 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
*divergentless*functions.

[4] Namely, that

[6] This expansion is often written as

where

and

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This page was prepared and is maintained by R. Victor Jones,

Last updated April 11, 2000