(a) A particle is free to move only in the \(x\) direction, subject to the potential energy \(U=U_{0} e^{-\alpha x^{2}}\), where \(\alpha\) and \(U_{0}\) are positive constants. Sketch constant-Hamiltonian curves in a phase diagram, including values of \(H\) with \(HU_{0}\).

(b) Repeat part (a) if \(U_{0}<0\) and \(\alpha>0\), for values of \(H\) including those with \(0>H>U_{0}, H=0\), and \(H>0\).