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 Soviet physicist Lev Landau. 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 Soviet physicist Lev Landau. Consider a two-dimensional system of non-interacting particles with charge q and spin S confined to an area A = LxLy in the x-y plane. Apply a uniform magnetic field B = ( 0 0 B ) {displaystyle mathbf {B} ={egin{pmatrix}0\0\Bend{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 There is some gauge freedom in the choice of vector potential for a given magnetic field. The Hamiltonian is gauge invariant, which means that adding the gradient of a scalar field to  changes the overall phase of the wave function by an amount corresponding to the scalar field. But physical properties are not influenced by the specific choice of gauge. For simplicity in calculation, choose the Landau gauge, which is where B=|B| and x̂ is the x component of the position operator.