[UPSC CSE 2008] Schrodinger’s Wave equation and applications (Q204, 30M)

The Hamiltonian of a particle moving along the x-axis is given by \hat{H} = -\alpha \frac{d^2}{dx^2} + 16\alpha \hat{x}^2, where \alpha is a real and positive constant having dimensions of energy. (i) If \psi(x) = A e^{-2x^2}, find the normalization constant A. Check whether \psi is an eigen function of \hat{H}. If yes, find the corresponding eigen value. (ii) Calculate the probability of finding the particle anywhere along the negative x-axis. (iii) Find the eigen value of \hat{H} corresponding to the eigen function \phi(x) = x\psi(x), where \psi(x) is the same as in part (i). (iv) Are the wave functions \psi(x) and \phi(x) orthogonal ?