= Solution
The proposed potentials give $\nabla\times A=(0,0,B)$ and $-\nabla\phi=(0,E,0)$. Substituting $\psi=e^{ikx}e^{-i\mathcal Et/\hbar}\varphi(y)$ gives
$$
\left[-\frac{\hbar^2}{2m}\frac{d^2}{dy^2}
+\frac{(\hbar k-eBy)^2}{2m}+eEy\right]\varphi
=\mathcal E\varphi.
$$
This is a harmonic oscillator with cyclotron frequency and centre
$$
\omega_1=\frac{|eB|}{m},
\qquad
y_0=\frac{\hbar k}{eB}-\frac{mE}{eB^2}
$$
for $eB>0$ as written. Completing the square yields
$$
\mathcal E=\hbar\omega_1\left(n+\frac12\right)+W,
\qquad
W=\frac{\hbar kE}{B}-\frac{mE^2}{2B^2}.
$$
\b[Thus the <Landau levels in crossed electric and magnetic fields> retain their oscillator spacing but acquire a linear dependence on $k$, lifting their guiding-centre degeneracy.]
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