The ground-state wave function of a one-dimensional simple harmonic oscillator of mass \(m\) and force-constant \(k\) is the Gaussian function \(\psi(x)=A e^{-\alpha x^{2}}\) where \(A\) is a normalization constant (adjusted to make \(\int \psi^{*} \psi d x=1\) ) and \(\alpha\) is also a constant.

(a) Using Schrödinger's equation, find \(\alpha\) in terms of \(m, k\), and \(\hbar \equiv h / 2 \pi\).

(b) Find the energy eigenvalue for this wave function, in terms of the same constants.

(c) Compare with the energy predicted for this ground state by the "old quantum theory."