Landau quantization

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Landau quantization in quantum mechanics is a description of how a charged particle in a magnetic field can only take discrete energy values. It is named after the Soviet physicist Lev Landau.

1 Derivation
2 See also
3 Further reading

[edit] Derivation

Consider a two-dimensional electron system confined to the x-y plane with an applied magnetic field B perpendicular to that plane — that is, along the z-axis.

The Hamiltonian of this system is

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

where e represents the electron charge, $\hat{\mathbf{p}}=\frac{\hbar}{i}\nabla$ is the momentum operator, and A the (magnetic) vector potential, which is related to the magnetic field by

$B=\nabla\times A.$

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

$A= \begin{pmatrix}0\\Bx\\0 \end{pmatrix}$ ,

where $B=|\mathbf{B}|$ . In this gauge the Hamiltonian is

$H=\frac{1}{2m}\left(\hat{p}_x^2+\hat{p}_y^2-2eBx\hat{p}_y+(eB)^2x^2\right)$ .

The next step is to look for solutions of the time-independent Schrödinger equation: $H Ψ(x, y) = E Ψ(x, y)$ . Since $\hat{p}_y$ commutes with the Hamiltonian, the eigenfunctions, $Ψ$ , of the Hamiltonian can be expressed as

$\Psi(x,y)=e^{ik_y y} \phi(x)$ .

Substituting this equation into the Schrödinger equation gives

$\frac{1}{2m}\left[-\hbar^2\frac{d^2}{dx^2}+(eB)^2\left(x-\frac{\hbar k_y}{eB}\right)^2\right]\phi(x)=E\phi(x)$ .

This is the well known Schrödinger equation for the quantum harmonic oscillator shifted in coordinate space by $x_0=\frac{\hbar k_y}{eB}$ . The solutions are therefore $φ(x) = ψ n (x - x 0)$ , where $ψ n$ are solutions of the quantum harmonic oscillator (see Quantum harmonic oscillator.)

[edit] Landau levels

From comparison with the known energies of a quantum harmonic oscillator, we obtain the eigenvalues

$E_n=\hbar\omega\left(n+\frac{1}{2}\right),\quad n\geq 0$

with the cyclotron frequency

$\omega=\frac{eB}{m}.$

The levels labeled by n are called Landau levels.

[edit] Discussion

This derivation treats x and y as slightly asymmetric. Because of the symmetry of the system, however, there is no physical quantity which differentiates these coordinates. The same result could have been obtained with an appropriate exchange of x and y.

[edit] Degeneracy

The state of the electron is characterized by the quantum numbers $k y$ and n. Since for any given energy $E n$ there are many possible values of $k y$ , there is the possibility of Landau level degeneracy. If the dimension size in the y direction is $L y$ , and periodic boundary conditions are assumed, $k y$ can take the values $k_y=\frac{2\pi}{L_y}r$ , where r is a non-negative integer. The allowed values of r are further restricted by the condition that the center of the oscillator, $x 0$ , has to lie within the limits $L_x\geq x_0\geq 0$ . This gives the upper bound

$0\leq r\leq\frac{eB}{2\pi\hbar}L_xL_y=r_\textrm{max}$ ,

which is the maximum number of electrons that can occupy a given Landau level n.

So with increasing field B, more electrons can fit into a given Landau level. Because of Zeeman splitting, each Landau level splits in two for spin up/spin down respectively. $r max$ is the number for just one spin species.

[edit] 3D-system

The above derivation assumed an electron confined in the z-direction, which is a relevant experimental situation — found in two-dimensional electron gases, for instance. This assumption is not essential for the obtained results. If electrons are free to move along the z direction, the wave function acquires an additional multiplicative term $e^{ik_z z}$ , and the energy corresponding to this free motion, $\frac{\hbar^2 k_z^2}{2m}$ , has to be added to E. In any case, the motion in the x-y-plane, perpendicular to the magnetic field, is quantized.