Electric-field displacement of a quantum harmonic oscillator ($x_0=qE/k$)

For $H=p^2/(2m)+k{x}^2/2-qEx$, completing the square translates the well by $x_0=qE/k$ and shifts every energy by $-q^2{E}^2/(2k)$. The wavefunction of the ground state is translated without changing its width. Consequently the induced electric dipole moment is exactly linear in $E$, even when a low-order perturbative approximation to the state is inaccurate.

Let $x_{\mathrm{zp}}=\sqrt{\hslash /(2m\omega )}$ and $\eta =x_0/(2x_{\mathrm{zp}})$. The normalized first-order truncation is $(|0⟩+\eta |1⟩)/\sqrt{1+{\eta }^2}$. Its mean displacement is $2\eta x_{\mathrm{zp}}/(1+{\eta }^2)$, agreeing with $x_0$ only at small $|\eta |$. The exact translated ground state is a coherent state whose expansion contains all number states and has mean occupation ${\eta }^2$. Correct linear response does not justify extrapolating a truncated state to large fields.