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{{Unreferenced|date=July 2009}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
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>
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


== Wave function ==
Registered users will be able to choose between the following three rendering modes:


Using [[polar coordinates]] on the 1-dimensional ring, the [[wave function]] depends only on the [[angle|angular]] [[coordinate]], and so
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:<math> \nabla^2 = \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} </math>
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^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
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


:<math> \int_{0}^{2 \pi} \left| \psi ( \theta ) \right|^2 \, d\theta = 1\ </math>,
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


and
==Demos==


:<math> \ \psi (\theta) = \ \psi ( \theta + 2\pi)</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


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>
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


== Energy eigenvalues ==
==Test pages ==


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
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


:<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
*[[Inputtypes|Inputtypes (private Wikis only)]]
:<math> e^{\pm i 2 \pi \frac{r}{\hbar} \sqrt{2 m E}  } = 1 = e^{i 2 \pi n}</math>
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
The eigenfunction and eigenenergies are
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
:<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 2n+1 states with energies up to 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 (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 (molecule)|buckyballs]]" (buckminsterfullerene). These molecules are exceptionally stable.
 
The above explains why the 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)</math> electrons, as electrons have additionally two possible orientations of their spins.
 
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]].
 
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]]
 
{{DEFAULTSORT:Particle In A Ring}}
[[Category:Quantum models]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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