Problem 6.89

A particle of mass \(m\) is located in a spherically symmetrical potential well \(U(r)=0\) for \(r< r_{0}\) and \(\tilde{U}(r)=U_{0}\) for \(r>r_{0}\) (a) By means of the substitution \(\psi(r)=\chi(r) / r\) find the equation defining the proper values of energy \(E\) of the particle for \(E< U_{0}\), when its motion is described by a wave function \(\psi(r)\) depending only on \(r .\) Reduce that equation to the form \[ \sin k r_{0}=\pm k r_{0} \sqrt{\hbar^{2} / 2 m r_{0}^{2} U_{0}}, \text { where } k=\sqrt{2 m E} / \hbar \] (b) Calculate the value of the quantity \(r_{0}^{2} U_{0}\) at which the first level appears.