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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
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| :<math> -\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi </math>
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| == Wave function ==
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| Using [[polar coordinates]] on the 1 dimensional ring, the [[wave function]] depends only on the [[angle|angular]] [[coordinate]], and so
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| :<math> \nabla^2 = \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} </math>
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| 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
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| :<math> \int_{0}^{2 \pi} \left| \psi ( \theta ) \right|^2 \, d\theta = 1\ </math>, | |
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| and
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| :<math> \ \psi (\theta) = \ \psi ( \theta + 2\pi)</math>
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| Under these conditions, the solution to the Schrödinger equation is given by
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| :<math> \psi_{\pm}(\theta) = \frac{1}{\sqrt{2 \pi}}\, e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} \theta } </math>
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| == Energy eigenvalues ==
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| 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 | |
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| :<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
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| :<math> e^{\pm i 2 \pi \frac{r}{\hbar} \sqrt{2 m E} } = 1 = e^{i 2 \pi n}</math>
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| The eigenfunction and eigenenergies are
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| :<math> \psi(\theta) = \frac{1}{\sqrt{2 \pi }} \, e^{\pm i n \theta }</math>
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| :<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>
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| 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 2n+1 states with energies up to an energy indexed by the number n.
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| 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.
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| 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]].
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| This simple model can be used to find approximate energy levels of some ring molecules, such as benzene.
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| ==Application==
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| 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 "[[buckyballs]]" (buckminsterfullerene). These molecules are exceptionally stable.
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| The above explains why the ring behaves like a circular [[waveguide]], with the valence electrons orbiting in both directions.
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| To fill all energy levels up to n requires <math>2\times(2n+1)</math> electrons, as electrons have additionally two possible orientations of their spins.
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| The rule that <math>4n+2</math> excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is known as the [[Hückel's rule]].
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| Further in rotational spectroscopy this model may be used as an approximation of rotational energy levels.
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| == See also ==
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| * [[Angular momentum]]
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| * [[Harmonic analysis]]
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| * [[One-dimensional periodic case]]
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| {{DEFAULTSORT:Particle In A Ring}}
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| [[Category:Quantum models]]
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