= Solution
A simpler physical coordinate is $Y=y+p_xc/(eB)$. For $\sigma_3$ <eigenvalue> $s_z=\pm1$, the equation becomes an oscillator of <mass> $m$, <frequency> $\omega_c=|eB|/(mc)$, and auxiliary <energy>
$$
\varepsilon=\frac{E^2-m^2c^4-p_z^2c^2+e\hbar Bc\,s_z}{2mc^2}.
$$
If the source's variable $\xi=(p_xc+eBy)/(\hbar c)=b_B Y$ is used literally, $b_B=eB/(\hbar c)$, then the same canonical oscillator form has $\boxed{M=m/b_B^2,\ \omega=\omega_c}$ and the same $\varepsilon$. Because $\xi$ is an inverse length, this auxiliary $M$ is not a physical particle <mass> with ordinary <Mass dimension>[mass dimensions]; the rescaling must be carried through both kinetic and potential terms.
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