When a particle with charge is moving in a homogeneous magnetic field, quantum mechanics decrees that its allowed energies split into degenerate discrete Landau levels, a phenomenon known as Landau quantization.
Starting from the Hamiltonian for a particle with mass in a vector potential :
We choose , yielding a magnetic field pointing in the -direction with strength . The Hamiltonian becomes:
The only position operator occurring in is , so . Because appears in an unmodified kinetic energy term, and the corresponding does not occur at all, the particle has completely free motion in the -direction. Likewise, because does not occur in , we can replace by its eigenvalue , although the motion is not free, due to .
Based on the absence of and , we make the following ansatz for the wavefunction : a plane wave in the and directions, multiplied by an unknown :
Inserting this into the time-independent Schrödinger equation gives, after dividing out the plane wave exponential :
By defining the cyclotron frequency and rearranging, we can turn this into a 1D quantum harmonic oscillator in , with a couple of extra terms:
The potential minimum is shifted by , and a plane wave in contributes to the energy . In any case, the energy levels of this type of system are well-known:
And is then as follows, where is the known quantum harmonic oscillator solution:
Note that this wave function contains (also inside ), but is absent from the energy . This implies degeneracy: assuming periodic boundary conditions , then can take values of the form , for .
However, also occurs in the definition of , so the degeneracy is finite, since must still lie inside the system, or, more formally, :
Isolating this for , we find the following upper bound of the degeneracy:
Where is the area of the confinement in the -plane. Evidently, the degeneracy of each level increases with larger , but since , the energy gap between each level increases too. In other words: the density of states is a constant with respect to the energy, but the states get distributed across the differently depending on .
- L.E. Ballentine, Quantum mechanics: a modern development, 2nd edition, World Scientific.