[UPSC CSE 1990] Schrodinger’s Wave equation and applications (Q236, 60M)

Set up Schrodinger's equation for a one dimensional harmonic oscillator. Assuming that the solution is of the from \psi = A e^{-\alpha x^2} f(x) where \alpha is a constant and f(x) is a polynomial in x, find the energy eigen values and the ground state eigenfunction. Comment on the relation between the ground slate energy and Heisenberg's uncertainty principle.