Section 22.10 Motion on a Sphere
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 (19.5.17)–(19.5.18) and, called spherical harmonics, will occur whenever one solves a partial differential equation that involves spherical symmetry.
Following the techniques in Section 22.4, 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 (19.5.18), with solutions exactly as we found them for the ring in Section 22.5. The other equation will be (19.5.17) with slightly different labels for the unknown constant, i.e.
\begin{equation}
\left({\sin\theta}\frac{\partial}{\partial\theta}
\left({\sin\theta}\frac{\partial}{\partial\theta}\right)
- A\sin^2\theta - m^2\right) P(\theta) = 0\tag{22.10.1}
\end{equation}
where \(m\) is a known integer and \(A\) is an unknown constant.
\begin{equation}
\frac{\partial^2 P}{\partial z^2}
- \frac{2z}{1-z^2}\frac{\partial P}{\partial z}
- \frac{A}{1-z^2} P-\frac{m^2}{(1-z^2)^2} P
= 0\tag{22.10.2}
\end{equation}
This equation is the Associated Legendre’s Equation. For any given integer value of \(m\text{,}\) it is a Sturm-Liouville equation
[cross-reference to target(s) "sturm" missing or not unique]. 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 [cross-reference to target(s) "legass" missing or not unique] 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
[cross-reference to target(s) "odepower" missing or not unique]. One example in this chapter [cross-reference to target(s) "odeser" missing or not unique] 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 [cross-reference to target(s) "eigenexpand" missing or not unique].
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.
\begin{align}
\ket{\ell,m}\amp \doteq Y_\ell^m(\theta,\phi)\tag{22.10.3}\\
\amp = (-1)^{(m+|m|)/2}
\sqrt{\frac{(2\ell+1)}{4\pi}
\frac{(\ell-\absm)!}{(\ell+\absm)!}}
\> P_\ell^m(\cos\theta) \> e^{im\phi}\tag{22.10.4}
\end{align}
See
[cross-reference to target(s) "sphhar2" missing or not unique] for more details about the algebra and [cross-reference to target(s) "sphhar" missing or not unique] for some interactive visualizations.
