   Next: Quantization of an electromagnetic Up: A Brief Introduction to Previous: Gauges

# Quantization of an electromagnetic field with standing waves

Maxwell's equations for a source free environment are (3.1) (3.2) (3.3)

In this environment, and The simplicity of these four equations begs for even further simplification, whereby we introduce the vector potential, There are many ways to fit these equations while maintaining the validity of the Maxwell equations. The coulomb potential suffices and is preferrable due to its simplicity: , . The identity coupled with our previous assertation and equation 3 above yield (3.4)

We will quantize this centralized magnetic potential. To completely specify the field we would have to describe its values for all points in space; it is customary to develop the quantization in a theoretical cube, and then let the volume of the cube expand to infinity to accomplish a full discription.

Our derivation can consider standing or plane waves. The case of standing waves is quicker. We assume the magnetic potential has a solution of form (3.5)

Under this seperation of variables, (3.6) (3.7)

Being in the standing wave regime, there can be no currents on the boundary, implying that and . But of course, we have already doomed ourselves to the fact that . These consequences become important as follows.

The energy stored by our electromagnetic field is (3.8)

We assume via our target solution that our spatial modes will have the orthogonality (3.9)

Using this orthogonality, we have (3.10)

The rightmost integral can be taken with an algebraic manipulation. We examine it as follows: A snap shot of the latter yields: This suggest the generating equation Thus our vector identity is More appropriate for our needs, we rearrange the previous equation to find: Reidentifying and , we have Remembering that this is under a volume integral, we quickly see that the first term Stokes away as a surface integral which our previously established boundary conditions founded as zero. The second term is taken care of by the use of a previous vector identity. Catching up, we have (3.11)

With equation 7 and our boundary conditions, this becomes, (3.12)

This form is fully equivalant to the harmonic oscillator, and with the following set: (3.13) (3.14) (3.15)

we complete the standing wave quantization by concluding that (3.16)   Next: Quantization of an electromagnetic Up: A Brief Introduction to Previous: Gauges
Timothy Jones 2006-05-30