We will now relax the restriction that the mass be confined to the ring and, instead, let it range over the surface of a sphere of radius \(r_0\text{.}\) The results of this analysis yield predictions that can be successfully compared with experiment for molecules and nuclei that rotate more than they vibrate. For this reason, the problem of a mass confined to a sphere is often called the rigid rotor problem. Furthermore, the solutions that we will find for (6.5.17)–(6.5.18) and, called spherical harmonics, will occur whenever one solves a partial differential equation that involves spherical symmetry.
Following the techniques in Section 14.1, you should be able to write down the Schrödinger equation for a particle restricted to a sphere and use the separation of variables procedure to obtain an equivalent set of ordinary differential equations. One of the equations you obtain will be (6.5.18), with solutions exactly as we found them for the ring in Section 14.2. The other equation will be (6.5.17) with slightly different labels for the unknown constant, i.e.
This equation is the Associated Legendre’s Equation. For any given integer value of \(m\text{,}\) it is a Sturm-Liouville equation Section 5.1. The solutions are called Associated Legendre Functions. When the eigenvalue \(A\) takes the special form \(A=-\ell(\ell+1)\) and for \(|m|\le \ell\text{,}\) the solutions \(P_{\ell}^m(z)\) form a basis for any sufficiently smooth function on the interval \(-1\le z\le 1\) that does not blow up at the endpoints. More information about Associated Legendre functions can be found in Section 15.3 and, of course, online.
If you recognize a known Sturm-Liouville equation, you can always just look up the solutions, called special functions. They arise from particular geometric situations, so knowing where the equation comes from will help with guessing which equation it is. If you don’t recognize the solution, then you can solve it with power series methods Chapter 7. One example in this chapter Section 7.2 uses the power series method to solve Legendre’s equation (the special case \(m=0\)). To use the solutions, called Legendre polynomials, to make an eigenfunction expansion (a generalization of Fourier series) see multiple sections in Chapter 5.
A complete basis of eigenstates for a quantum particle confined to the surface of a sphere (or for the angular part of any PDE involving the Laplacian in spherical coordinates) can be found from multiplying any basis state \(\Phi_m(\phi)\) with any basis state \(P_{\ell}^m(\cos\theta)\text{,}\) subject to the restrictions \(\ell\) is a non-negative integer, \(m\) is an integer (positive, negative, or zero), and \(m\le \ell\text{.}\) Properly normalized, these functions are called spherical harmonics.
