= Solution
For a particle of charge $q$ and <mass> $m$ in a uniform <magnetic field> $\mathbf B=B\hat z$, <minimal coupling> replaces <momentum> by $\mathbf p-q\mathbf A$. Choose <Landau gauge> $\mathbf A=(0,Bx,0)$ and first restrict motion to the $xy$ plane. Since $p_y=\hbar k_y$ is conserved,
$$
H=\frac{p_x^2}{2m}+\frac{(\hbar k_y-qBx)^2}{2m}
=\frac{p_x^2}{2m}+\frac12m\omega_c^2(x-x_c)^2,
$$
$$
\omega_c=\frac{|qB|}{m},\qquad x_c=\frac{\hbar k_y}{qB},\qquad
\boxed{E_n=\hbar\omega_c(n+\tfrac12),\quad n=0,1,2,\ldots.}
$$
These <Landau levels> quantize the classical cyclotron motion. The <wavefunctions> are harmonic-oscillator functions centred at $x_c$, multiplied by $e^{ik_yy}$. Changing $k_y$ shifts the <guiding centre> without changing the <energy>. Gauge-dependent labels such as $k_y$ are not themselves observables; the level spacing and degeneracy are gauge independent. The <magnetic length> $\ell_B=\sqrt{\hbar/|qB|}$ is the transverse orbital scale.
For a large rectangle of sides $L_x,L_y$, periodicity along $y$ gives $\Delta k_y=2\pi/L_y$. Requiring centres inside the sample gives, for each orbital level and each specified <spin> state,
$$
\boxed{D=\frac{L_xL_y}{2\pi\ell_B^2}
=\frac{|qB|A}{h}=\frac{|\Phi|}{h/|q|}.}
$$
Thus each <magnetic flux quantum> supplies one guiding-centre state. The continuum bulk count neglects boundary corrections; magnetic periodic conditions on a torus require integer flux count. A finite confining edge lifts the simple flat guiding-centre <spectrum> near that edge.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2015/paper-1-landau-levels.png]
{title=Translated oscillator centres and degenerate orbital Landau levels in a uniform field}
<spin> adds a Zeeman term. For <electrons> with gyromagnetic factor $g$, its contribution is $g\mu_B Bm_s$, with $m_s=\pm1/2$ and $\mu_B=e\hbar/(2m_e)$. If $g=2$ and the orbital <mass> is $m_e$, the branches become $\hbar\omega_c n$ and $\hbar\omega_c(n+1)$: the lowest combined level has one <spin> branch, while positive levels coincide in pairs. In a material, effective <mass> and effective $g$ need not give that coincidence.
In three dimensions, free motion parallel to $B$ adds $\hbar^2k_z^2/(2m)$, so each orbital index labels a one-dimensional continuum rather than an isolated three-dimensional <energy>. Resolved magnetic quantization requires the spacing to exceed substantial thermal and disorder broadening. The changing occupation of these levels underlies magnetic quantum oscillations; in two dimensions their degeneracy and filling are central to the <integer quantum Hall effect>. These statements assume nonrelativistic, approximately independent particles, rather than arbitrary interacting or disordered spectra.
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