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| {{Refimprove|date=February 2013}}
| | Historical past of the of the author is generally Gabrielle Lattimer. Fish bearing is something her hubby doesn't really like however she does. Idaho is where her home is normally and she will undoubtedly move. Software developing is what she may but she's always wanted her own business. She are running and maintaining a blog here: http://circuspartypanama.com<br><br>my blog :: [http://circuspartypanama.com clash of clans cheat gems] |
| {{Disputed|Multiple problems in the introduction|date = March 2011}}
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| 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.
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| 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.)
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| 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.
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| In [[special relativity|relativistic]] [[quantum field theory]], a stable bound state of {{mvar|n}} particles with masses {{bigmath|''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 frame|center of mass energy]] which is less than {{bigmath|''m''<sub>1</sub> + … + ''m''<sub>''n''</sub> }}. An [[unstable]] bound state (see [[resonance (particle physics)|resonance]]) shows up as a pole with a [[complex number|complex]] center of mass energy.
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| ==Examples==
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| [[Image:Particle overview.svg|thumb|400px|An overview of the various families of elementary and composite particles, and the theories describing their interactions]]
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| * 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]].
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| * A [[Atomic nucleus|nucleus]] is a bound state of [[proton]]s and [[neutron]]s ([[nucleon]]s).
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| * A [[positronium]] "atom" is an [[resonance|unstable bound state]] of an [[electron]] and a [[positron]]. It decays into [[photon]]s.
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| * The [[proton]] itself is a bound state of three [[quark]]s (two [[up quark|up]] and one [[down quark|down]]; one [[color charge|red]], one [[color charge|green]] and one [[color charge|blue]]). However, unlike the case of the hydrogen atom, the individual quarks can never be isolated. See [[color confinement|confinement]].
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| * The [[eigenstates]] of the [[Hubbard model]] and [[Jaynes-Cummings-Hubbard model]] (JCH) Hamiltonian in the two-excitation subspace are also examples of bound states. In Hubbard model, two repulsive [[bosonic]] [[atoms]] can form a bound pair in an [[optical lattice]].<ref>
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| {{cite journal
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| | author = K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. H. Denschlag, A. J. Daley, A. Kantian, H. P. Buchler and P. Zoller
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| | title = Repulsively bound atom pairs in an optical lattice
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| | journal = [[Nature]]
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| | year = 2006
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| | volume = 441
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| | pages = 853
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| |arxiv = cond-mat/0605196 |bibcode = 2006Natur.441..853W |doi = 10.1038/nature04918 }}
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| </ref><ref>
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| {{cite journal
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| | title = Dimer of two bosons in a one-dimensional optical lattice
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| | author = Javanainen, Juha and Odong, Otim and Sanders, Jerome C.
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| | journal = [[Phys. Rev. A]]
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| | volume = 81
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| | issue = 4
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| | pages = 043609
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| | numpages = 12
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| |date=Apr 2010
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| | doi = 10.1103/PhysRevA.81.043609
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| | url = http://link.aps.org/doi/10.1103/PhysRevA.81.043609
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| | publisher = [[American Physical Society]]
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| |arxiv = 1004.5118 |bibcode = 2010PhRvA..81d3609J }}
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| </ref><ref> | |
| {{cite journal
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| | author = M. Valiente and D. Petrosyan
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| | title = Two-particle states in the Hubbard model
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| | journal = [[J. Phys. B]]: At. Mol. Opt. Phys.
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| | year = 2008
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| | volume = 41
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| | pages = 161002
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| }}
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| </ref> The JCH Hamiltonian also supports two-[[polariton]] bound states when the photon-atom interaction is sufficiently strong. In particular, the two polaritons associated with the bound states exhibit a strong [[correlation]] such that they stay close to each other in [[position space]]. The results discussed has been published in Ref.<ref>
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| {{cite journal
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| | title = Two-polariton bound states in the Jaynes-Cummings-Hubbard model
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| | author = Max T. C. Wong and C. K. Law
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| | journal = [[Phys. Rev. A]]
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| | volume = 83
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| | issue = 5
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| | pages = 055802
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| | numpages = 4
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| |date=May 2011
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| | doi = 10.1103/PhysRevA.83.055802
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| | url = http://link.aps.org/doi/10.1103/PhysRevA.83.055802
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| | publisher = [[American Physical Society]]
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| |arxiv = 1101.1366 |bibcode = 2011PhRvA..83e5802W }}
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| </ref>
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| ==In mathematical quantum physics==
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| Let {{mvar|H}} 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 {{mvar|H}} and <math>\rho = \rho(t_0) </math> be a statistical operator on {{mvar|H}}. Let {{mvar|A}} be an [[observable]] on {{mvar|H}} and let <math>\mu(A,\rho)</math> be the induced probability distribution of {{mvar|A}} with respect to {{mvar|ρ}} on the [[Borel set|Borel σ-algebra]] on <math>\mathbb{R}</math>. Then the evolution of {{mvar|ρ}} induced by {{mvar|U}} is said to be '''bound''' with respect to {{mvar|A}} 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>.
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| '''Example:'''
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| Let <math>H = L^2(\mathbb{R}) </math> and let {{mvar|A}} be the position observable. Let <math>\rho = \rho(0) \in H</math> have compact support and <math>[-1,1] \subseteq \mathrm{Supp}(\rho)</math>.
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| * If the state evolution of {{mvar|ρ}} "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 {{mvar|ρ}} is not a bound state with respect to the position.
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| * 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.
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| * More generally: If the state evolution of {{mvar|ρ}} "just moves {{mvar|ρ}} inside a bounded domain", then {{mvar|ρ}} is also a bound state with respect to position.
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| ==See also==
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| *[[Composite field]]
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| *[[Resonance]]
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| *[[Bethe–Salpeter equation]]
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| ==References==
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| {{Reflist|2}}
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| {{Particles}}
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| {{Chemical bonds}}
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| {{DEFAULTSORT:Bound State}}
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum field theory]]
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Historical past of the of the author is generally Gabrielle Lattimer. Fish bearing is something her hubby doesn't really like however she does. Idaho is where her home is normally and she will undoubtedly move. Software developing is what she may but she's always wanted her own business. She are running and maintaining a blog here: http://circuspartypanama.com
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