 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 $p_y=\hslash k_y$ is conserved,These Landau levels quantize the classical cyclotron motion. The wavefunctions are harmonic-oscillator functions centred at $x_c$, multiplied by $e^{i{k}_{y}y}$. 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 ${\ell }_B=\sqrt{\hslash /|qB|}$ 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,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.

spin adds a Zeeman term. For electrons with gyromagnetic factor $g$, its contribution is $g{\mu }_{B}B{m}_s$, with $m_s=\pm 1/2$ and ${\mu }_B=e\hslash /(2m_e)$. If $g=2$ and the orbital mass is $m_e$, the branches become $\hslash {\omega }_{c}n$ and $\hslash {\omega }_c(n+1)$: 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 ${\hslash }^2{k}_z^2/(2m)$, 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.