Editing Particle in a ring

{{Short description|Concept in quantum mechanics}}
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 (physics)|configuration space]] is the [[circle]] <math>S^1</math>) is

:<math> -\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi </math>

== 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>

== 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.

== Application ==

In [[organic chemistry]], [[aromatic]] compounds contain atomic rings, such as [[benzene]] rings (the [[Friedrich August Kekulé von Stradonitz|Kekulé]] structure) consisting  of five or six, usually [[carbon]], atoms. So does the surface of "[[buckyball]]s" (buckminsterfullerene). This ring behaves like a circular [[waveguide]], with the valence electrons orbiting in both directions. To fill all energy levels up to n requires <math>2\times(2n+1)=4n+2</math> electrons, as electrons have additionally two possible orientations of their spins. This gives exceptional stability ("aromatic"), and is known as the [[Hückel's rule]].

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

== See also ==

* [[Angular momentum]]
* [[Harmonic analysis]]
* [[One-dimensional periodic case]]
* [[Semicircular potential well]]
* [[Spherical potential well]]

== References ==
{{reflist}}


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[[Category:Quantum models]]