Landau quantization

Landau quantization in quantum mechanics is the quantization of the cyclotron orbits of charged particles in magnetic fields. As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels. The Landau levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. Landau quantization is directly responsible for oscillations in electronic properties of materials as a function of the applied magnetic field. It is named after the Jewish Soviet physicist Lev Landau.

Consider a two-dimensional system of non-interacting particles with charge q and spin S confined to an area A = L_xL_y in the x-y plane.

Apply a uniform magnetic field ${\displaystyle \mathbf {B} ={\begin{pmatrix}0\\0\\B\end{pmatrix}}}$ along the z-axis. In CGS units, the Hamiltonian of this system is

Here, p̂ is the canonical momentum operator and  is the electromagnetic vector potential, which is related to the magnetic field by

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