Schrödinger wave equation — UPSC IFoS 2018 Physics PYQ

Write down the Schrödinger wave equation for a one-dimensional harmonic oscillator in which a particle of mass m and frequency \omega is subject to a parabolic potential V(x) = m\omega^2 x^2/2. Let one of the possible energy eigen states be given by \psi(x) = Ax e^{-x^2/x_0^2}. Find the energy E corresponding to the eigen state given by \psi(x). Is it a ground state energy or one of the excited state energies ? Comment.