One of the problem’s assigned in Foundations of QM asked us to consider the number of modes of the ground state function of a wave bouncing around inside of a cubical microwave cavity. The answer was three, and to me this seemed interesting, what if it was a tetrahedron shape? and so on… For a tetrahedron I think the ground state mode would be the one between the two opposing points of the pyramid but then if the E-field is going back and forth the long way wouldn’t the B-field be compressed at the points of the pyramid?
During my visit at the LTC today, Dr Noe and I were talking about the number of modes inside of a cavity, and he mentioned someone who had been taking copper shapes, banging them up so they would be heavily dented and then bouncing a wave around inside of it to carefully see when modes would form.
This lead to talking about how waves are composed of waves in various harmonic modes. For example, if sin(x) is the ground mode and sin(2x) is the first excited state, you can sum the two creating a new wave (one with some localization, I think) This summation can be analyzed using a Fourier Analysis, by asking, what is the area under the composite wave? And then adding and subtracting in the area’s provided by the additional modes.