Landau Quantization - Derivation

Derivation

Consider a two-dimensional system of non-interacting particles with charge and spin confined to an area in the x-y plane. Apply a uniform magnetic field along the z-axis. Using CGS units, the Hamiltonian of this system is

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

There is some freedom in the choice of vector potential for a given magnetic field. However, the Hamiltonian is gauge invariant, which means that adding the gradient of a scalar field to A changes the overall phase of the wave function by an amount corresponding to the scalar field. Physical properties are not influenced by the specific choice of gauge. For simplicity in calculation, choose the Landau gauge, which is

where and is the x component of the position operator. In this gauge the Hamiltonian is

The operator commutes with this Hamiltonian since the operator is absent due to the choice of gauge. Then the operator can be replaced by its eigenvalue . The Hamiltonian can also be written more simply by noting that the cyclotron frequency is, giving

This is exactly the Hamiltonian for the quantum harmonic oscillator, except shifted in coordinate space by .

To find the energies, note that translating the harmonic oscillator potential left or right does not change the energies. The energies of this system are identical to those of the quantum harmonic oscillator:

The energy does not depend on the quantum number, so there will be degeneracies.

For the wave functions, recall that commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the direction and harmonic oscillator eigenstates shifted by an amount in the direction:

In sum, the state of the electron is characterized by two quantum numbers, and .