Editing Particle in a ring (section)

== Energy eigenvalues ==

The [[energy]] [[eigenvalue]]s <math> E </math> are [[quantization (physics)|quantize]]d because of the periodic [[boundary condition]]s, 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>

The eigenfunction and eigenenergies are
:<math> \psi(\theta) = \frac{1}{\sqrt{2 \pi}} \, e^{\pm i n \theta }</math>
:<math> E_n = \frac{n^2 \hbar^2}{2 m R^2} </math> where <math>n = 0,\pm 1,\pm 2,\pm 3, \ldots</math>

Therefore, there are two degenerate [[quantum state]]s for every value of <math> n>0 </math> (corresponding to <math> \ e^{\pm i n \theta}</math>). Therefore, there are <math>2n+1</math> states with energies up to an energy indexed by the number <math>n</math>.

The case of a particle in a one-dimensional ring is an instructive example when studying the [[quantization (physics)|quantization]] of [[angular momentum]] for, say, an [[electron]] orbiting the [[Atomic nucleus|nucleus]]. The [[azimuth]]al wave functions in that case are identical to the energy [[eigenfunction]]s of the particle on a ring.

The statement that any wavefunction for the particle on a ring can be written as a [[quantum superposition|superposition]] of [[energy]] [[eigenfunction]]s is exactly identical to the [[Fourier theorem]] about the development of any periodic [[function (mathematics)|function]] in a [[Fourier series]].

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