Problem 6.90

The wave function of a particle of mass \(m\) in a unidimensional potential field \(U(x)=k x^{2} / 2\) has in the ground state the form \(\psi(x)=A \mathrm{e}^{-\alpha x^{2}},\) where \(A\) is a normalization factor and \(\alpha\) is a positive constant. Making use of the Schrödinger equation, find the constant \(\alpha\) and the energy \(E\) of the particle in this state.