 Solution

Set $d=qE$. With the stated unit mass and oscillator frequency, complete the square: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 hasIts normalized eigenfunctions are the original oscillator eigenfunctions evaluated at $x-d$, in particular the ground state is $(\pi \hslash {)}^{-1/4}{e}^{-(x-d{)}^2/(2\hslash )}$. The energy gaps remain $\hslash$.