Particle in a ring
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In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle <math>S^1<math>) is
- <math> -\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi <math>
Using polar coordinates on the 1 dimensional ring, the wave function depends only on the angular coordinate, and so
- <math> \nabla^2 = \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} <math>
Requiring that the wave function be periodic in <math> \ \theta <math> with a period <math> 2 \ \pi<math> (from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions
- <math> \int_{0}^{2 \pi} \left| \psi ( \theta ) \right|^2 \, d\theta = 1\ <math>,
and
- <math> \ \psi (\theta) = \ \psi ( \theta + 2\ \pi)<math>
Under these conditions, the solution to the Schrodinger equation is given by
- <math> \psi(\theta) = \frac{1}{\sqrt{2 \pi}}\, e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} \theta } <math>
The energy eigenvalues <math> E <math> are quantized because of the periodic boundary conditions, and they are required to satisfy
- <math> e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} \theta } = e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} (\theta +2 \pi)}<math>, or
- <math> e^{\pm i 2 \pi \frac{r}{\hbar} \sqrt{2 m E} } = 1 = e^{i 2 \pi n}<math>
This leads to the energy eigenvalues
- <math> E = \frac{n^2 \hbar^2}{2 m r^2} <math> where <math>n = 0,1,2,3, \ldots<math>
The full wave functions are, therefore
- <math> \psi(\theta) = \frac{1}{\sqrt{2 \pi}} \, e^{\pm i n \theta }<math>
Except for the case <math>n = 0<math>, there are two quantum states for every value of <math> n <math> (corresponding to <math> \ e^{\pm i n \theta}<math>). Therefore there are 2n+1 states with energies less than an energy indexed by the number n.
The case of a particle in a one-dimensional ring is an instructive example when studying the quantization of angular momentum for, say, an electron orbiting the nucleus. The azimuthal wave functions in that case are identical to the energy eigenfunctions of the particle on a ring.
Interestingly, the statement that any wavefunction for the particle on a ring can be written as a superposition of energy eigenfunctions is exactly identical to Fourier's theorem about the development of any periodic function in a Fourier series.
See also
One-dimensional periodic case, Angular momentum, Harmonic analysis.