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.