Editing Particle in a ring (section)

== Wave function ==
[[Image:Quantum-rigid-rotator 1+2-animation-color.gif|thumb|Animated wave function of a “coherent” state consisting of eigenstates n=1 and n=2.]]
Using [[polar coordinates]] on the 1-dimensional ring of radius R, the [[wave function]] depends only on the [[angle|angular]] [[coordinate]], and so<ref>{{cite book |last=Cox |first=Heater |title=Problems and Solutions to accompany Physical Chemistry: a Molecular Approach |publisher=University Science Books |page=141 |isbn=978-0935702439}}</ref>
:<math> \nabla^2 = \frac{1}{R^2} \frac{\partial^2}{\partial \theta^2} </math>

Requiring that the wave function be [[periodic function|periodic]] in <math> \ \theta </math> with a period <math> 2 \pi</math> (from the demand that the wave functions be single-valued [[function (mathematics)|function]]s on the [[circle]]), and that they be [[normalizing constant|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 Schrödinger equation is given by

:<math> \psi_{\pm}(\theta) = \frac{1}{\sqrt{2 \pi  }}\, e^{\pm i \frac{R}{\hbar} \sqrt{2 m E} \, \theta } </math>