Inquantum mechanics, the case of aparticle in a one-dimensional ring is similar to theparticle in a box. TheSchrödinger equation for afree particle which is restricted to a ring (technically, whoseconfiguration space is thecircle${\displaystyle S^1}$) is

${\displaystyle -\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\psi =E\psi }$

with boundary conditions

${\displaystyle \psi (\theta +2\pi )=\psi (\theta )}$

expressing the fact that the particle is in a ring.

Usingpolar coordinates on the 1-dimensional ring of radius R, thewave function depends only on theangularcoordinate, and so^[1]

${\displaystyle {\mathrm{\nabla }}^2=\frac{1}{R^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^2}}$

Requiring that the wave function beperiodic in${\displaystyle \theta }$ with a period${\displaystyle 2\pi }$ (from the demand that the wave functions be single-valuedfunctions on thecircle), and that they benormalized leads to the conditions

${\displaystyle {\int }_0^{2\pi }{\left|\psi (\theta )\right|}^{2}d\theta =1}$,

and

${\displaystyle \psi (\theta )=\psi (\theta +2\pi )}$

Under these conditions, the solution to the Schrödinger equation is given by

${\displaystyle {\psi }_{\pm }(\theta )=\frac{1}{\sqrt{2\pi }}{e}^{\pm i\frac{R}{\hslash }\sqrt{2mE}\theta }}$

Theenergyeigenvalues${\displaystyle E}$ arequantized because of the periodicboundary conditions, and they are required to satisfy

${\displaystyle e^{\pm i\frac{R}{\hslash }\sqrt{2mE}\theta }=e^{\pm i\frac{R}{\hslash }\sqrt{2mE}(\theta +2\pi )}}$ or

${\displaystyle e^{\pm i2\pi \frac{R}{\hslash }\sqrt{2mE}}=1=e^{i2\pi n}}$

The eigenfunction and eigenenergies are

${\displaystyle \psi (\theta )=\frac{1}{\sqrt{2\pi }}{e}^{\pm in\theta }}$

${\displaystyle E_n=\frac{n^2{\hslash }^2}{2m{R}^2}}$ where${\displaystyle n=0,\pm 1,\pm 2,\pm 3,\dots }$

Therefore, there are two degeneratequantum states for every value of${\displaystyle n>0}$ (corresponding to${\displaystyle e^{\pm in\theta }}$). Therefore, there are${\displaystyle 2n+1}$ states with energies up to an energy indexed by the number${\displaystyle n}$.

The case of a particle in a one-dimensional ring is an instructive example when studying thequantization ofangular momentum for, say, anelectron orbiting thenucleus. Theazimuthal wave functions in that case are identical to the energyeigenfunctions of the particle on a ring.

The statement that any wavefunction for the particle on a ring can be written as asuperposition ofenergyeigenfunctions is exactly identical to theFourier theorem about the development of any periodicfunction in aFourier series.

This simple model can be used to find approximate energy levels of some ring molecules, such as benzene.

Inorganic chemistry,aromatic compounds contain atomic rings, such asbenzene rings (theKekulé structure) consisting of five or six, usuallycarbon, atoms. So does the surface of "buckyballs" (buckminsterfullerene). This ring behaves like a circularwaveguide, with the valence electrons orbiting in both directions. To fill all energy levels up to n requires${\displaystyle 2\times (2n+1)=4n+2}$ electrons, as electrons have additionally two possible orientations of their spins. This gives exceptional stability ("aromatic"), and is known as theHückel's rule.

Further in rotational spectroscopy this model may be used as an approximation of rotational energy levels.

• Angular momentum
• Harmonic analysis
• One-dimensional periodic case
• Semicircular potential well
• Spherical potential well

• Quantum models

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