# Energy eigenfunctions of particle on a ring

Let's say we have a free particle of mass $\mu$ that is constrained to move on a ring of radius $a$. Let the ring lie in the $xy$-plane, and let the $z$-axis go through the centre $O$ of the ring, which we also take to be the origin of coordinates.

Since the potential energy $V = 0$ (the particle is free), the energy is purely kinetic. The Hamiltonian is $$H = \frac{L_z^2}{2 I}$$ where $I = \mu a^2$ is the moment of inertia with respect to the origin.

I want to find the energy eigenfunctions for the system and write a general expression for the solution of the time-dependent Schrödinger equation. Can I say that, in this case, the wave function $\Psi(\mathbf{r}, t)$ depends only only the angular variables $\phi, \theta$, if we work in spherical coordinates? Should I separate variables, and write $$\Psi(\mathbf{r}, t) = \psi(\theta, \phi) \exp(-i Et / \hbar)$$ to solve this problem?

• On a ring there's only one angular coordinate. Otherwise, what you're doing is fine. – Javier Mar 29 '16 at 22:28
• So, I should use the angle $\phi$ as the only variable, then write $L_z = -i \hbar \frac{ \partial}{\partial \phi}$, and solve the equation $H \psi(\phi) = \frac{-i \hbar^2}{2 I} \frac{ \partial^2}{ \partial \phi^2} \psi(\phi) = E \psi(\phi)$ for the energy? – Kamil Mar 29 '16 at 22:36
• A quantum particle cannot be free if it is constrained to be on a ring. You could formulate a quantum problem for a particle that is constrained to an annular ring (of width w and radius a). Here the potential would be $V=0$ for the region $r=a+or- w/2$ and $V=infinity$ elsewhere. – Lewis Miller Mar 30 '16 at 0:41