Charged Particle in a Magnetic Field: Landau Levels

A charged particle in a uniform magnetic field is the simplest system in which gauge structure, massive degeneracy, and topology all appear at once. Its quantized energy levels — Landau levels — underlie the quantum Hall effects, Shubnikov–de Haas oscillations, and much of the physics of electrons in solids. The Aharonov–Bohm effect it exhibits shows that the vector potential, not just the field, is physically significant in quantum mechanics.

1. Minimal Coupling

Electromagnetic fields enter the Hamiltonian by minimal coupling: replace the canonical momentum with the kinetic momentum $\hat{\mathbf{\pi }}=\hat{\mathbf{p}}-q\mathbf{A}(\hat{\mathbf{r}})$,

$$\hat{H}=\frac{1}{2m}(\hat{\mathbf{p}}-q\mathbf{A}{)}^2+q\phi ,$$

where $\mathbf{A}$ is the vector potential ($\mathbf{B}=\nabla \times \mathbf{A}$). This is the same prescription that, promoted to a local symmetry, generates the gauge interactions of QFT. Unlike $\hat{\mathbf{p}}$, the kinetic-momentum components do not commute in a field:

$$[{\hat{\pi }}_x,{\hat{\pi }}_y]=i\hslash q{B}_z.$$

2. Landau Levels

For a uniform field $\mathbf{B}=B\hat{z}$, the commutator above is exactly that of ladder operators: the transverse motion is a harmonic oscillator in disguise. Its spectrum is the set of Landau levels

$$\boxed{E_n=\hslash {\omega }_c(n+\frac{1}{2}),{\omega }_c=\frac{qB}{m},n=0,1,2,\dots }$$

with ${\omega }_c$ the cyclotron frequency — the quantum echo of classical circular orbits. (Adding free motion along $\hat{z}$ contributes a continuous ${\hslash }^2{k}_z^2/2m$.) Each level is macroscopically degenerate: the number of states per unit area is $B/{\Phi }_0$ with ${\Phi }_0=h/q$ the flux quantum, since the orbit's guiding center can sit anywhere. This huge degeneracy — many states at exactly the same energy — is what makes partially filled Landau levels a playground for strong-correlation physics.

3. Gauge Choice

The vector potential for a uniform field is not unique: the Landau gauge $\mathbf{A}=(0,Bx,0)$ and the symmetric gauge $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{r}$ describe the same field. They give different-looking wavefunctions (plane waves in $y$ vs. circularly symmetric states) but identical spectra and physics — a concrete illustration of gauge invariance: observables depend on $\mathbf{B}$, not on the choice of $\mathbf{A}$. Choosing a gauge is like choosing coordinates.

4. The Aharonov–Bohm Effect

Gauge freedom does not mean $\mathbf{A}$ is unphysical. In the Aharonov–Bohm effect, electrons pass on either side of a confined magnetic flux through a region where $\mathbf{B}=0$ but $\mathbf{A}\ne 0$. Each path acquires a phase $\frac{q}{\hslash }\int \mathbf{A}\cdot d\mathbf{l}$, and the interference pattern shifts by

$$\Delta \varphi =\frac{q}{\hslash }\oint \mathbf{A}\cdot d\mathbf{l}=\frac{q\Phi }{\hslash },$$

depending only on the enclosed flux $\Phi$ — even though the electrons never enter the field. This shows the electromagnetic potential has direct physical consequences in quantum mechanics (the phase is gauge-invariant because it is a closed loop), a topological effect and an early signal of the geometric-phase structure underlying gauge theory.

See also

• Harmonic oscillator — the algebra reused for Landau levels.
• Symmetries & conservation — gauge symmetry and generators.
• QFT: QED from postulates — minimal coupling as local gauge symmetry.