Hamilton’s equations — UPSC IFoS 2024 Physics PYQ

Consider a particle of mass m and charge q moving under the influence of a one-dimensional harmonic oscillator potential. Assume that it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 - qEx Derive the energy expression and wave function of the n\text{th} excited state.