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{{Unreferenced|date=December 2009}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
{{Disputed|Multiple problems in the introduction|date = March 2011}}


In [[physics]], a '''bound state''' describes a system where a [[particle]] is subject to a [[Potential Energy|potential]] such that the particle has a tendency to remain localised in one or more regions of space. The potential may be either an external potential, or may be the result of the presence of another particle.
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.


In [[quantum mechanics]] (where the number of particles is conserved), a bound state is a state in [[Hilbert space]] that corresponds to two or more particles whose [[interaction energy]] is less than the total energy of each separate particle, and therefore these particles cannot be separated unless [[energy]] is spent. The [[energy spectrum]] of a bound state is discrete, unlike the continuous spectrum of isolated particles. (Actually, it is possible to have unstable bound states with a positive interaction energy provided that there is an "energy barrier" that has to be [[quantum tunnelling|tunnelled]] through in order to decay. This is true for some [[Radionuclide|radioactive nuclei]] and for some [[electret]] materials able to carry electric charge for rather long periods.)
Registered users will be able to choose between the following three rendering modes:


In general, a stable bound state is said to exist in a given potential of some dimension if stationary wavefunctions exist (normalized in the range of the potential). The energies of these wavefunctions are negative.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


In [[theory of relativity|relativistic]] [[quantum field theory]], a stable bound state of n particles with masses m<sub>1</sub>, ..., m<sub>n</sub> shows up as a [[pole (complex analysis)|pole]] in the [[S-matrix]] with a center of mass energy which is less than m<sub>1</sub>+...+m<sub>n</sub>. An [[unstable]] bound state (see [[resonance]]) shows up as a pole with a [[complex number|complex]] center of mass energy.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==Examples==
'''source'''
[[Image:Particle overview.svg|thumb|400px|An overview of the various families of elementary and composite particles, and the theories describing their interactions]]
:<math forcemathmode="source">E=mc^2</math> -->
* A [[proton]] and an [[electron]] can move separately; the total center-of-mass energy is positive, and such a pair of particles can be described as an ionized atom. Once the electron starts to "orbit" the proton, the energy becomes negative, and a bound state – namely the [[hydrogen atom]] – is formed. Only the lowest energy bound state, the [[ground state]] is stable. The other [[excited state]]s are unstable and will decay into bound states with less energy by emitting a [[photon]].
* A [[Atomic nucleus|nucleus]] is a bound state of [[proton]]s and [[neutron]]s ([[nucleon]]s).
* A [[positronium]] "atom" is an [[resonance|unstable bound state]] of an [[electron]] and a [[positron]]. It decays into [[photon]]s.
* The [[proton]] itself is a bound state of three [[quark]]s (two [[up quark|up]] and one [[down quark|down]]; one [[Quantum chromodynamics|red]], one [[Quantum chromodynamics|green]] and one [[Quantum chromodynamics|blue]]). However, unlike the case of the hydrogen atom, the individual quarks can never be isolated. See [[color confinement|confinement]].


==In mathematical quantum physics==
<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].
Let <math>H</math> be a complex separable Hilbert space, <math> U = \lbrace U(t) \mid t \in \mathbb{R} \rbrace </math> be a one-parametric group of unitary operators on <math> H </math> and <math>\rho = \rho(t_0) </math> be a statistical operator on <math>H</math>. Let <math>A</math> be an observable on <math>H</math> and let <math>\mu(A,\rho)</math> be the induced probability distribution of <math>A</math> with respect to <math>\rho</math> on the Borel <math>\sigma</math>-algebra on <math>\mathbb{R}</math>. Then the evolution of <math>\rho</math> induced by <math>U</math> is said to be '''bound''' with respect to <math>A</math> if <math>\lim_{R \rightarrow \infty} \sum_{t \geq t_0} \mu(A,\rho(t))(\mathbb{R}_{> R}) = 0 </math>, where <math>\mathbb{R}_{>R} = \lbrace x \in \mathbb{R} \mid x > R \rbrace </math>.  


'''Example:'''
==Demos==
Let <math>H = L^2(\mathbb{R}) </math> and let <math>A</math> be the position observable. Let <math>\rho = \rho(0) \in H</math> have compact support and <math>[-1,1] \subseteq \mathrm{Supp}(\rho)</math>.


* If the state evolution of <math>\rho</math> "moves this wave package constantly to the right", e.g. if <math>[t-1,t+1] \in \mathrm{Supp}(\rho(t)) </math> for all <math>t \geq 0</math>, then <math>\rho</math> is not a bound state with respect to the position.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


* If <math>\rho</math> does not change in time, i.e. <math>\rho(t) = \rho</math> for all <math>t \geq 0</math>, then <math>\rho</math> is a bound state with respect to position.


* More generally: If the state evolution of <math>\rho</math> "just moves <math>\rho</math> inside a bounded domain", then <math>\rho</math> is also a bound state with respect to position.
* 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.


==See also==
==Test pages ==
*[[Composite field]]
*[[Resonance]]
*[[Bethe-Salpeter equation]]


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


{{DEFAULTSORT:Bound State}}
*[[Inputtypes|Inputtypes (private Wikis only)]]
[[Category:Quantum mechanics]]
*[[Url2Image|Url2Image (private Wikis only)]]
[[Category:Quantum field theory]]
==Bug reporting==
 
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 .
[[ca:Partícula composta]]
[[cy:Cyflwr rhwym]]
[[de:Gebundener Zustand]]
[[et:Liitosakesed]]
[[es:Partícula compuesta]]
[[fr:État lié]]
[[ja:束縛状態]]
[[pt:Partícula composta]]
[[ur:حالت پیوند]]

Latest revision as of 23:52, 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


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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