A charge in a uniform magnetic field circles at the cyclotron frequency $\omega_{c}=|q|B/m$, and classically any radius, so any energy, is allowed. Quantum mechanically the motion across the field is an oscillator in disguise. The two kinetic momenta across the field have the commutator $i\hbar qB$, just like a position and a momentum; the Hamiltonian is the sum of their squares; so the energies are those of an oscillator, $\hbar\omega_{c}(n+\tfrac12)$, with free motion along the field added. These are the Landau levels. Each is hugely degenerate: the energy does not depend on where the orbit is centred, and one state fits in each area that holds a flux quantum $h/|q|$. Filling them one after another gives the quantum Hall effect.

A theatre whose rows are the energies: every row is very long, with one seat for every flux quantum through the floor. Turn up the field and the rows move apart and every row gains seats.

The first four Landau levels, $\hbar\omega_{c}(n+\tfrac12)$, against the field , in units where at : each rises in proportion to , and so does the spacing between them. Every level also holds states per unit area, so a stronger field packs more states into each.

For a charge $q$ in a uniform field $B$ along $z$, with the cyclotron frequency $\omega_{c}=|q|B/m$, the energies are

• Each level holds $|q|B/h$ states per unit area across the field: one for every flux quantum $h/|q|$ through the sample.
• In the Landau gauge the states are $e^{i(p_{y}y+p_{z}z)/\hbar}\psi_{n}(x-x_{0})$, oscillator states centred at : a whole line of them for each .

Across the field the two kinetic momenta, scaled to have the commutator $i$, play the parts of an oscillator’s position and momentum. Their complex combination is a lowering operator with $[\hat{a},\hat{a}^{\dagger}]=1$, and the sum of their squares is : the Hamiltonian is an oscillator’s, with the spectrum of the oscillator chapter. Take ; for exchange and .

$m\hbar\omega_{c}=\hbar qB$, so the commutator $i\hbar qB$ becomes $i$.

$\hat{\Pi}_{z}=\hat{p}_{z}$ when has no component.

$[\hat{a},\hat{a}^{\dagger}]=\tfrac12\bigl(-i[\hat{u}_{x},\hat{u}_{y}]+i[\hat{u}_{y},\hat{u}_{x}]\bigr)=1$.

$2\hat{a}^{\dagger}\hat{a}=\hat{u}_{x}^{2}+\hat{u}_{y}^{2}+i[\hat{u}_{x},\hat{u}_{y}]$, and the commutator is .

$\hat{a}^{\dagger}\hat{a}$ has the eigenvalues $n=0,1,2,\dots$, as for the oscillator.

Across the field the motion is an oscillator at the cyclotron frequency $\omega_{c}=|q|B/m$, so the levels are $\hbar\omega_{c}(n+\tfrac12)$.

The two kinetic momenta across the field have a constant commutator, like a position and a momentum, and the Hamiltonian is the sum of their squares.

The energy does not depend on where an orbit is centred. Each level holds $|q|B/h$ states per unit area: one for each flux quantum.

The number of states in a level is the flux through the sample divided by the flux quantum $h/|q|$.

Great work! You now know how to:

• turn the motion across a field into an oscillator
• find the Landau levels and their spacing
• count the states in each level