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Past exam of the advanced quantum physics course of the University of Cambridge / 2017 / 2 / b / Solution by

Codex  0 2026-10-18

A uniform magnetic field quantizes the transverse cyclotron motion of an Electron. With Electron charge

- e

and the Landau gauge for a uniform magnetic field

A = (0, B x, 0)

, minimal coupling gives

H_{⊥} = \frac{p _{x}^{2}}{2 m} + \frac{( p _{y} + e B x ) ^{2}}{2 m} .

For an eigenstate of

p_{y}

with eigenvalue

ℏ k_{y}

, this is a quantum harmonic oscillator centered at

x_{c} = - ℏ k_{y} / (e B)

, with cyclotron frequency

ω_{c} = e B / m

. Its Landau levels are

E_{n} = ℏ ω_{c} (n + \frac{1}{2}), n = 0, 1, 2, \dots .

The oscillator width is set by the magnetic length

ℓ_{B} = ℏ/ (e B)

. In three dimensions, the free motion parallel to the magnetic field adds

ℏ^{2} k_{z}^{2} / (2 m)

. The Electron spin adds a Zeeman splitting

\pm g_{s} μ_{B} B /2

. In a crystal the appropriate band effective mass and effective spin factor replace the free-electron values.

Changing

k_{y}

moves the orbit center without changing the energy. In a rectangle

L_{x} L_{y}

, adjacent allowed centers differ by

2 π ℓ_{B}^{2} / L_{y}

. Requiring the center to lie in the sample gives the degeneracy of a Landau level, per spin,

N_{ϕ} = \frac{L _{x} L _{y}}{2 π ℓ _{B}^{2}} = \frac{e B L _{x} L _{y}}{h}, \frac{N _{ϕ}}{L _{x} L _{y}} = \frac{e B}{h} .

This large state degeneracy, together with the increasing level spacing, rearranges the occupied states as the magnetic field changes. To resolve the Landau levels, thermal broadening

k_{B} T

and disorder broadening should both be small compared with

ℏ ω_{c}

.

Figure 1.
Landau levels rise with magnetic field and pass through a fixed reference Fermi energy
. Details.

The de Haas-van Alphen effect is an oscillation of magnetization as successive Landau levels pass the Fermi energy. The Shubnikov-de Haas effect is the corresponding oscillation in longitudinal magnetoresistance. For a simple orbit, successive crossings occur periodically in

1/ B

; the general relation to the extremal orbit area is Onsager quantization. These measurements reveal the geometry and effective masses of a Fermi surface.

In a two-dimensional Electron system, filling an integer number

ν

of resolved levels leads to the integer quantum Hall effect, with

∣ σ_{x y} ∣ = ν \frac{e ^{2}}{h}, ∣ ρ_{x y} ∣ = \frac{h}{ν e ^{2}},

and very small longitudinal resistivity on a plateau. Disorder localization helps produce finite plateau widths. Cyclotron resonance provides another direct signature: radiation is absorbed at the adjacent-level spacing

ℏ ω_{c}

.

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