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Sep1-10, 03:36 PM
P: 527
Well, in general, the quantum mechanical eigenfunctions do not correspond to classical orbits. These eigenfunctions of the Hamiltonian form a basis of the Hilbert space -- which is one reason why we work with them -- but when we take the classical limit we usually do not find that these eigenfunctions turn into classical states. One reason you pointed out -- more or less. The eigenfunctions do not always posses the symmetries of the classical action (in this case these wavefunctions break rotational symmetry). In this case the origin of this "missing" symmetry is the fact that one has to make a choice of gauge in order to solve for the eigenfunctions. Choosing a gauge will in general break one or more symmetries (you can work in the symmetric gauge which is rotationally symmetric, but then you will lose translational symmetry).

We can, however, construct an wavefunction which is "as classical as possible". These wavefunctions are called coherent states. These states localize the particle as much as possible in both coordinate and momentum space. They are constructed as some linear combination of all the eigenstates. If you want to know what these coherent states exactly look like in the QHE you should look up the notes by S.M Girvin.

One thing I should add is that these eigenfunctions in some sense correspond to a whole collection of classical orbits with a radius of roughly the magnetic length.