= Solution
A <magnetic field> quantizes the transverse cyclotron motion into <Landau levels>. For a parabolic band, this is a harmonic oscillator with $E_n=\hbar\omega_c(n+1/2)$, $n=0,1,\ldots$, where $\omega_c=eB/m_c$ and the half arises from its zero-point energy. Equivalently, semiclassical orbit quantization gives the stated flux sequence in increments of $h/e$.
To relate real and momentum space, take $\mathbf B=B\hat{\mathbf z}$ and electron charge $-e$. The semiclassical <Lorentz force> is
$$
\hbar\dot{\mathbf k}=-e\mathbf v\times\mathbf B.
$$
Consequently $dx=\hbar\,dk_y/(eB)$ and $dy=-\hbar\,dk_x/(eB)$. The real-space orbit is a rotated copy of the transverse wavevector orbit, scaled by $\hbar/(eB)$. Their areas satisfy $S_r=[\hbar/(eB)]^2S_k$. Since $BS_r=(n+1/2)h/e$,
$$
\boxed{S_{k,n}=\frac{2\pi eB}{\hbar}\left(n+\frac12\right).}
$$
This is <Onsager quantization> for the ordinary half-integer phase. For a circular parabolic orbit, $S_k=\pi k^2$ reproduces the Landau energy spacing. The flux increment $h/e$ here concerns a single electron; it is distinct from the superconducting pair flux unit.
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