= Solution
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> $\mathbf A=(0,Bx,0)$, <minimal coupling> gives
$$
H_\perp=\frac{p_x^2}{2m}+\frac{(p_y+eBx)^2}{2m}.
$$
For an <eigenstate> of $p_y$ with <eigenvalue> $\hbar k_y$, this is a <quantum harmonic oscillator> centered at $x_c=-\hbar k_y/(eB)$, with <cyclotron frequency> $\omega_c=eB/m$. Its <Landau levels> are
$$
\boxed{E_n=\hbar\omega_c\left(n+\frac12\right),\qquad n=0,1,2,\ldots}.
$$
The oscillator width is set by the <magnetic length> $\ell_B=\sqrt{\hbar/(eB)}$. In three dimensions, the free motion parallel to the <magnetic field> adds $\hbar^2k_z^2/(2m)$. The <Electron> <spin> adds a <Zeeman splitting> $\pm g_s\mu_BB/2$. 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 $L_xL_y$, adjacent allowed centers differ by $2\pi\ell_B^2/L_y$. Requiring the center to lie in the sample gives the <degeneracy of a Landau level>, per <spin>,
$$
N_\phi=\frac{L_xL_y}{2\pi\ell_B^2}=\frac{eBL_xL_y}{h},\qquad
\boxed{\frac{N_\phi}{L_xL_y}=\frac{eB}{h}}.
$$
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 $k_BT$ and disorder broadening should both be small compared with $\hbar\omega_c$.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2017-landau-levels.png]
{title=Landau levels rise with magnetic field and pass through a fixed reference Fermi energy}
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 $\nu$ of resolved levels leads to the <integer quantum Hall effect>, with
$$
|\sigma_{xy}|=\nu\frac{e^2}{h},\qquad |\rho_{xy}|=\frac{h}{\nu e^2},
$$
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 $\hbar\omega_c$.
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