Chemistry 352

The particle on a ring is a simple but powerful quantum model. A particle of mass \( m \) is constrained to move on a circle of fixed radius \( R \). There is no potential energy along the ring—the particle is free to move—but its motion is restricted to a closed loop.

The position of the particle is described by a single angular coordinate \( \phi \), where

\[ 0 \le \phi < 2\pi. \]

Because the particle can move continuously around the ring, this system has perfect rotational symmetry. That symmetry will strongly constrain the allowed quantum states.

Cyclical boundary condition

The defining feature of the particle on a ring is its cyclical (periodic) boundary condition. A point at angle \( \phi \) is physically identical to the point at \( \phi + 2\pi \).

This means the wavefunction must satisfy

\[ \psi(\phi) = \psi(\phi + 2\pi). \]

Unlike the particle in a box, the wavefunction does not go to zero at the boundaries. Instead, it must join smoothly onto itself after one complete revolution. This single requirement is what quantizes the allowed states.

Schrödinger equation and eigenfunctions

With no potential energy, the time-independent Schrödinger equation becomes

\[ -\frac{\hbar^2}{2I}\frac{d^2\psi}{d\phi^2} = E\psi, \]

where \( I = mR^2 \) is the moment of inertia.

The solutions that satisfy the periodic boundary condition are

\[ \psi_m(\phi) = \frac{1}{\sqrt{2\pi}} e^{im\phi}, \qquad m = 0, \pm1, \pm2, \pm3, \ldots \]

The requirement that the wavefunction be single-valued forces the quantum number \( m \) to be an integer. This is a direct consequence of the cyclical boundary condition.

Energy levels and degeneracy

The energy eigenvalues are

\[ E_m = \frac{\hbar^2 m^2}{2I}. \]

Several important features follow immediately:

• The energies depend on \( m^2 \), not on the sign of \( m \).
• The state with \( m = 0 \) is nondegenerate.
• For every \( |m| \ge 1 \), the states \( +m \) and \( -m \) have the same energy.

Thus, all excited energy levels are doubly degenerate. This degeneracy reflects the fact that the particle can circulate around the ring with equal energy in either the clockwise or counterclockwise direction.

The degeneracy is a direct consequence of the rotational symmetry of the system. There is no preferred direction of motion on the ring.

Connection to angular momentum

The operator for angular momentum about the center of the ring is

\[ \hat{L}_z = -i\hbar \frac{d}{d\phi}. \]

The eigenfunctions \( \psi_m(\phi) \) are also eigenfunctions of \( \hat{L}_z \):

\[ \hat{L}_z \psi_m = m\hbar\,\psi_m. \]

The particle on a ring therefore provides one of the clearest examples of quantized angular momentum arising purely from boundary conditions and symmetry.

Key points (one glance)

• The coordinate \( \phi \) is periodic: \( \psi(\phi)=\psi(\phi+2\pi) \).
• Periodic boundary conditions force integer quantum numbers \( m \).
• Energies scale as \( m^2 \), not \( |m| \).
• The \( m=0 \) state is nondegenerate.
• All excited states are doubly degenerate (\( \pm m \)).
• Degeneracy reflects rotational symmetry and direction of motion.