Consider a charged particle in the one-dimensional harmonic oscillator potential. Suppose we turn on a weak electric field (E), so that the potential energy is shifted by an amount H′=−qEx.
(a) Show that there is no first-order change in the energy levels, and calculate the second-order correction. *Hint:* See Problem 3.39.
(b) The Schrödinger equation can be solved directly in this case, by a change of variables: x′≡x−(qE/mω2). Find the exact energies, and show that they are consistent with the perturbation theory approximation.