OMYGAWD! I can see that I'll have to give up soon.
The solutions to the wave equation inside the cavity have a real part ~0 in the exponent.
It's not a question of a Schroedingers Wave Equation, it's a question of Maxwell's Equations. *** Yeah I've got my old copy of Jackson. MTW too, for all the good it did me. Maxwell's equations can be used to form a wave equation too. I only bring up the Schroedinger equation because the solutions to simple particle-in-a-box examples are easy to generate and easy to visualize. Anyone who even began a decent Physics program should have done many of these. The probability function for the Quantum example *looks like* the amplitude for the EM example...analogy is a good tool.
The boundary condition at the inside surface of the copper box splices together the solutions in the cavity and inside the conductor.
*** What conductor? The shell is equipotential unless you're trying to play head games with me so there follows there can be no current flow through it except radialy to the outside of the sphere. *** DC, Yes. AC, things are happening. To solve the diffeq's for an EM wave incident on a conducting surface you have to make the solutions !inside! the conductor match the solutions outside the conductor. Only if the conductor is *perfect* does your assumption of nothing going on inside the conductor make sense. BTW - the skin depth for Cu at 100MHz is about 0.00026". The skin depth is proportional to f^(-0.5). In brief, here's what happens to the wave incident on a copper sheet: Wave is incident on surface Most of the wave is reflected the better the conductor, the more is reflected the portion that is not reflected is attenuated in the conductor ( loss ) any amplitude on the other surface of the sheet radiates energy. It should be obvious that, if the conductor is good, there will be very little amplititude inside the conductor, low loss reflection. Further, that 4 mils of Cu should provide ~80 dB loss in even the tiny amount that is not reflected at the inner surface. At 100MHz. I haven't solved this system for many years and I'm not inclined to go back to it now. Take my word for it : a little RF can get through a copper sheet but only a *very* little. It's the finite conductivity that alters the simple scenario. *** Let's walk through it using your model.... The spark gap generates sparks and that builds up free electrons in the space inside the sphere (whether it is gas filled or a vacuum is irrelevant). As that charge builds up it will be all of one type, electrons. Now the electrons repel each other and therefor move in a circular motion with the spark gap as the center. They strike the surface of the sphere and tunnel through to the outside surface where they reside. The amount of charge at any one point is related to the curvature of the surface at that point. Since a sphere is constant curvature the charge will be evenly distributed. It will continue to build up so long as you supply power to the spark gap. In an ideal world it will get bigger and bigger. In the real world at some point insulation breaks down and normal current flow takes place. *** Wild! Forget all this, this patchwork of pieces doesn't hold together as a description of the physical problem. If it helps, forget about the spark gap. Waves waves waves. Start with a wave packet in the box. Don't worry about how it got there. Worry about keeping it there. Best regards Jim, Mike