8: Central Potentials

In this chapter, we shall investigate the interaction of a non-relativistic particle of mass \(m\) and energy \(E\) with various so-called central potentials, \(V(r)\), where \(r=(x^{\,2}+y^{\,2}+z^{\,2})^{1/2}\) is the radial distance from the origin. It is, of course, most convenient to work in spherical coordinates—\(r\), \(\theta\), \(\phi\)—during such an investigation. (See Section [s8.3].) Thus, we shall be searching for stationary wavefunctions, \(\psi(r,\theta,\phi)\), that satisfy the time-independent Schrödinger equation (see Section [sstat]) \[\label{e9.1} H\,\psi = E\,\psi,\] where the Hamiltonian takes the standard non-relativistic form \[\label{e9.2} H = \frac{p^{\,2}}{2\,m} + V(r).\]