We are going to solve the same plasma in a constant B field in a different gauge.
This is the same as the 1D harmonic oscillator equation with and .
So we get the same energies with a much simpler calculation. The resulting states are somewhat strange and are not analogous to the classical solutions. (Note that an electron could be circulating about any field line so there are many possible states, just in case you are worrying about the choice of and and counting states.)
Jim Branson 2013-04-22