Hamilton’s equations — UPSC CSE 2023 Physics PYQ

Consider a particle of mass m and charge q moving under the influence of a one dimensional harmonic oscillator potential. Assume 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. Obtain the energy expression and the wave function of the n\text{th} excited state of the particle.