Charged oscillator in a uniform electric field

physics

Problem

A particle of mass mm and charge qq moves in a 1-D harmonic potential plus a uniform static electric field E\mathcal{E}: H=p22m+12mω2x2−qExH = \dfrac{p^2}{2m} + \tfrac{1}{2}m\omega^2 x^2 - q\mathcal{E}x. (a) Find the exact eigenvalues and eigenstates. (b) Treating H′=−qExH' = -q\mathcal{E}x as a perturbation, compute the first- and second-order energy corrections and show they reproduce the exact answer (all higher orders vanish).

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