All Problems Wave Properties of particles. Schrodinger Equation.
Problem 6.87
Find the possible values of energy of a particle of mass \(m\) located in a spherically symmetrical potential well \(U(r)=0\) for \(r< r_{0}\) and \(U(r)=\infty\) for \(r=r_{0},\) in the case when the motion of the particle is described by a wave function \(\psi(r)\) de pending only on \(r\) Instruction. When solving the Schrödinger equation, make the substitution \(\psi(r)=\chi(r) / r\)
6.87. Utilizing the substitution indicated, we get \[ \chi^{\prime \prime}+k^{2} \chi=0, \quad \text { where } \quad k^{2}=2 m E / \hbar^{2} \] We shall seek the solution of this equation in the form \(\chi=\) \(=a \sin (k r+\alpha)\). From the finiteness of the wave function \(\psi\) at the point \(r=0\) it follows that \(\alpha=0 .\) Thus, \(\psi=(a / r) \sin k r .\) From the boundary condition \(\psi\left(r_{0}\right)=0\) we obtain \(k r_{0}=n \pi,\) where \[ n=1,2, \ldots \text { Hence, } E_{n}=n^{2} \pi^{2} \hbar^{2} / 2 m r_{0}^{2} \]