Q&A
Q: How can we go from quantum description to the classical description of particles in a magnetic field ?
A: The classical cyclotron motion is most closely approximated by a
coherent state, just like for the harmonic oscillators. Coherent states are most easily developed in symmetric gauge as discussed in problem 5 of pset 4. See also the discussion in the posted notes by Prof. Jaffe.
Q: Have we consisdered the two possible spins of electrons when we calculated the filling of the Landau levels?
A: Due to coupling to the magnetic field, different spin oreintations do not have the same energies. Thus when filling the Landau levels it is convenient to consider filling each spin orientation separately. What we discussed in lecture can be considered as the story for one spin component (e.g. spin up). So we fill one electron for each single particle state. The story for the other spin component is exctly parallel.
Q: Are there any observables that are gauge variant?
A: No physical observable is gauge variant. Otherwise the theory will not be self-consistent, since the same physical quantity will then have different answers if you choose to calculate it using different gauges. Of course in experiments, we only measure one answer.
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