[UPSC CSE 2012] Schrodinger’s Wave equation and applications (Q187, 4+8=12M)

Consider the one-dimensional wavefunction \psi(x)=Axe^{-kx}, (0\leq x\leq\infty;\,k>0)

(i) Calculate A so that \psi(x) is normalized.

(ii) Using Schrödingers equation find the potential V(x) and energy E for which \psi(x) is an eigenfunction. (Assume that as x \to \infty, V(x) \to 0).