PHYS598MAP Spring 2005 Blog

1. Quantization of the EM field. The procedure to turn the EM field into something qauntum mechanical is to first quantize the field and then second quantize the field. We use the Coulomb, or transverse, gauge for working with quantum EM fields (and in QED).

2. First quantization is purely classical, and just means writing down the fields as expansions in an othornormal basis set of functions. In class, we did this by considering the fields in free space and choosing a box as our quantization volume. We're going to care about the vector potential A when we deal with light interacting with an atom, so we write down the wave equation for A and solve it; E and B are just derivatives of A.

3. Then we write down the total energy in the field, and that's our classical Hamiltonian. Cleverly, we can invent real valued coordinates Q and P and show that this Hamiltonian is just a simple harmonic oscillator Hamiltonian in Q and P. The fields are made quantum mechanical (this is second quantization) by the usual procedure — the coordinates are made operators and we introduce raising and lowering operators. The Hamiltonian is then diagonal in the mode occupation basis, or photon number basis.

4. At this point, the vacuum energy shows up. This is real, and hard to understand. The vacuum energy has physical consequences, such as in the Casimir and Casimir-Polder effects.

Cheers,

Brian