Quote by conway
I understand what you're saying and it's a natural mistake for people to make. Yes, in the onedimensional potential well, the energy levels get farther and farther apart the more electrons you add. But for the 2d well, the geometry exactly compensates for this sparseness. Go to 3d and the the density of energy levels actually increases the more electrons you add. For practical sizes, you can consider it a continuum.

That
still doesn't work!
Look at as 3D standing wave rectangular waveguide. If you connect a spectrum analyzer to it, you'll see various
modes that can be sustained in in. Make it larger to get more modes in it, and you can still detect "ripples" in the spectrum signifying the location of each mode. In fact, if I have a good enough resolution (and spectrum analyzers nowadays have amazing resolutions as it is already), I can certainly detect such modes.
Note that this is just a consideration of the energy state. We haven't even looked at how one would get the band dispersion of an ordinary metal. How would you propose to get that our of such a model?
Zz.