= Solution
Set $d=qE$. With the stated unit mass and oscillator frequency, complete the square:
$$
U_0+U_1=\frac{x^2}{2}-qEx=\frac{(x-d)^2}{2}-\frac{d^2}{2}.
$$
Translation by $d$ preserves the spectrum of the <quantum harmonic oscillator>, and the last term is a constant energy shift. The <displaced quantum harmonic oscillator> therefore has
$$
\boxed{E_n=\hbar\left(n+\frac12\right)-\frac{q^2E^2}{2},\qquad n\ge0.}
$$
Its normalized eigenfunctions are the original oscillator eigenfunctions evaluated at $x-d$, in particular the <ground state> is $(\pi\hbar)^{-1/4}e^{-(x-d)^2/(2\hbar)}$. The energy gaps remain $\hbar$.
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