Question

Consider a charged particle bound in the harmonic oscillator potential V (x) =½mω²x². A weak electric field E is applied to the system such that the potential energy is shifted by an amount H'=-qEx.

(a) Calculate the energy levels of the perturbed system to second order in the small perturbation.
(b) Show that the perturbed system can be solved exactly by completing the square in the
Hamiltonian. Compare the exact energies with the perturbation results found in (a).