Electron in constant magnetic field - classical vs quantum

[grelade]

Hi,

Movement of an electron in constant magnetic field, according to semiclassical QM, give rise to Landau levels - a quantization of energy. Everything would be fine but i find it difficult to reconcile these findings with classical point of view in which Lorentz force is acting on moving electron.
So, classically an electron will be moving in circular orbits according to condition:
$$m v^2 / r = evB$$
We get continuous spectrum of circular orbits each with different energy E.

However when we're considering the same situation in QM (B field in z direction) we get a wave function of the electron like this (according to wikipedia and my own calculations):
$$\Psi (x,y) = e^{-i k_x x} H_n(\frac{y-x_0^2 k_x}{x_0}) e^{-(\frac{y-x_0^2 k_x}{\sqrt{2} x_0})^2}$$
,where $$x_0=\sqrt{\hbar/eB}$$.

This solution is manifestly not circular-symmetric. Which it should be according to classical point of view. So plotting $$|\Psi|^2$$ don't seems to me like having any resemblance to classical motion at all and i don't have any idea why?

 

[xepma]

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.