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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:

[tex]m v^2 / r = evB[/tex]

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):

[tex]\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}

[/tex]

,where [tex]x_0=\sqrt{\hbar/eB}[/tex].

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

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# Electron in constant magnetic field - classical vs quantum

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