Separation of Variables on a Ring

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Section 22.4 Separation of Variables on a Ring

To begin our study of the properties of the solutions of Schrödinger’s equation in cases with spherical symmetry, we first consider the simpler case of the motion of a quantum particle of mass \(\mu\) confined to move on a ring of constant radius \(r_0\text{.}\) As with classical orbits, let’s assume that the ring lies in the \((x,y)\)-plane, so that in spherical coordinates \(\theta=\frac{\pi}{2}=\hbox{const}\text{.}\) Then, since \(\Psi\) is independent of \(r\) and \(\theta\text{,}\) derivatives with respect to those variables give zero and Schrödinger’s equation from Section 22.3 reduces to

\begin{align} i\hbar\frac{\partial \Psi}{\partial t} \amp =H_{\rm op}\Psi\notag\\ \amp = -\frac{\hbar^2}{2\mu}\, \frac{1}{r_0^2}\,\frac{\partial^2}{\partial\phi^2}\Psi + U(r_0)\Psi\tag{22.4.1} \end{align}

Redoing the separation of variables procedure of Section 19.5, and assuming that \(\Psi=T(t)\Phi(\phi)\) only, we obtain the following separated ordinary differential equations

\begin{align} \frac{d^2\Phi}{d\phi^2} \amp= -\frac{2I}{\hbar^2}\left(E-U(r_0)\right)\Phi\tag{22.4.2}\\ \frac{dT}{dt} \amp= -\frac{i}{\hbar}\, ET\tag{22.4.3} \end{align}

where we have used the substitution \(\mu r_0^2=I\text{,}\) in which \(I\) would be the moment of inertia of a classical particle of mass \(\mu\) traveling in a ring of radius \(r_0\) about the center-of-mass.

Alternatively, we could have obtained equations (22.4.2) and (22.4.3) from the results of our original separation of variables procedure (19.5.9), (19.5.15), (19.5.17), (19.5.18), by restricting the variables \(r\) and \(\theta\) to the equator, noticing that the functions \(R\) and \(P\) are therefore constant, and by finding that equation (19.5.15) reduces to:

\begin{equation} A = \frac{2\mu}{\hbar^2} (E-U(r_0)) r_0^2\tag{22.4.4} \end{equation}

and equation (19.5.17) then reduces to:

\begin{equation} B = -\frac{2\mu}{\hbar^2} (E-U(r_0)) r_0^2\text{.}\tag{22.4.5} \end{equation}