Problem 6.85

A particle of mass \(m\) is located in a unidimensional potential field \(U(x)\) whose shape is shown in Fig. \(6.2,\) where \(U(0)=\infty\). Find:

(a) the equation defining the possible values of energy of the particle in the region \(E<U_{0}\); reduce that equation to the form \[ \sin k l=\pm k l \sqrt{\hbar^{2} / 2 m l^{2} U_{0}} \] where \(k=\sqrt{2 m E / \hbar}\). Solving this equation by graphical means, demonstrate that the possible values of energy of the particle form a discontinuous spectrum; (b) the minimum value of the quantity \(l^{2} U_{0}\) at which the first energy level appears in the region \(E<U_{0} .\) At what minimum value of \(l^{2} U_{0}\) does the \(n\) th level appear?