Categories: Physics, Quantum mechanics.

Landau quantization

When a particle with charge qq 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 H^\hat{H} for a particle with mass mm in a vector potential A(Q^)\vec{A}(\hat{Q}):

H^=12m(p^qA)2\begin{aligned} \hat{H} &= \frac{1}{2 m} \big( \hat{p} - q \vec{A} \big)^2 \end{aligned}

We choose A=(y^B,0,0)\vec{A} = (- \hat{y} B, 0, 0), yielding a magnetic field B=×A\vec{B} = \nabla \times \vec{A} pointing in the zz-direction with strength BB. The Hamiltonian becomes:

H^=(p^xqBy^)22m+p^y22m+p^z22m\begin{aligned} \hat{H} &= \frac{\big( \hat{p}_x - q B \hat{y} \big)^2}{2 m} + \frac{\hat{p}_y^2}{2 m} + \frac{\hat{p}_z^2}{2 m} \end{aligned}

The only position operator occurring in H^\hat{H} is y^\hat{y}, so [H^,p^x]=[H^,p^z]=0[\hat{H}, \hat{p}_x] = [\hat{H}, \hat{p}_z] = 0. Because p^z\hat{p}_z appears in an unmodified kinetic energy term, and the corresponding z^\hat{z} does not occur at all, the particle has completely free motion in the zz-direction. Likewise, because x^\hat{x} does not occur in H^\hat{H}, we can replace p^x\hat{p}_x by its eigenvalue kx\hbar k_x, although the motion is not free, due to qBy^q B \hat{y}.

Based on the absence of x^\hat{x} and z^\hat{z}, we make the following ansatz for the wavefunction Ψ\Psi: a plane wave in the xx and zz directions, multiplied by an unknown ϕ(y)\phi(y):

Ψ(x,y,z)=ϕ(y)exp(ikxx+ikzz)\begin{aligned} \Psi(x, y, z) = \phi(y) \exp(i k_x x + i k_z z) \end{aligned}

Inserting this into the time-independent Schrödinger equation gives, after dividing out the plane wave exponential exp(ikxx+ikzz)\exp(i k_x x + i k_z z):

Eϕ=12m((kxqBy)2+p^y2+2kz2)ϕ\begin{aligned} E \phi &= \frac{1}{2 m} \Big( (\hbar k_x - q B y)^2 + \hat{p}_y^2 + \hbar^2 k_z^2 \Big) \phi \end{aligned}

By defining the cyclotron frequency ωcqB/m\omega_c \equiv q B / m and rearranging, we can turn this into a 1D quantum harmonic oscillator in yy, with a couple of extra terms:

(E2kz22m)ϕ=(12mωc2(ykxmωc)2+p^y22m)ϕ\begin{aligned} \Big( E - \frac{\hbar^2 k_z^2}{2 m} \Big) \phi &= \bigg( \frac{1}{2} m \omega_c^2 \Big( y - \frac{\hbar k_x}{m \omega_c} \Big)^2 + \frac{\hat{p}_y^2}{2 m} \bigg) \phi \end{aligned}

The potential minimum is shifted by y0=kx/(mωc)y_0 = \hbar k_x / (m \omega_c), and a plane wave in zz contributes to the energy EE. In any case, the energy levels of this type of system are well-known:

En=ωc(n+12)+2kz22m\begin{aligned} \boxed{ E_n = \hbar \omega_c \Big(n + \frac{1}{2}\Big) + \frac{\hbar^2 k_z^2}{2 m} } \end{aligned}

And Ψn\Psi_n is then as follows, where ϕ\phi is the known quantum harmonic oscillator solution:

Ψn(x,y,z)=ϕn(yy0)exp(ikxx+ikzz)\begin{aligned} \Psi_n(x, y, z) = \phi_n(y - y_0) \exp(i k_x x + i k_z z) \end{aligned}

Note that this wave function contains kxk_x (also inside y0y_0), but kxk_x is absent from the energy EnE_n. This implies degeneracy: assuming periodic boundary conditions Ψ(x ⁣+ ⁣Lx)=Ψ(x)\Psi(x\!+\!L_x) = \Psi(x), then kxk_x can take values of the form 2πn/Lx2 \pi n / L_x, for nZn \in \mathbb{Z}.

However, kxk_x also occurs in the definition of y0y_0, so the degeneracy is finite, since y0y_0 must still lie inside the system, or, more formally, y0[0,Ly]y_0 \in [0, L_y]:

0y0=kxmωc=2πnqBLxLy\begin{aligned} 0 \le y_0 = \frac{\hbar k_x}{m \omega_c} = \frac{\hbar 2 \pi n}{q B L_x} \le L_y \end{aligned}

Isolating this for nn, we find the following upper bound of the degeneracy:

nqBLxLy2π=qBAh\begin{aligned} \boxed{ n \le \frac{q B L_x L_y}{2 \pi \hbar} = \frac{q B A}{h} } \end{aligned}

Where ALxLyA \equiv L_x L_y is the area of the confinement in the (x,y)(x,y)-plane. Evidently, the degeneracy of each level increases with larger BB, but since ωc=qB/m\omega_c = q B / m, 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 EnE_n differently depending on BB.


  1. L.E. Ballentine, Quantum mechanics: a modern development, 2nd edition, World Scientific.