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2014-08-12T15:05:42Z

TommyFXDOkc:

{{Unreferenced|date=December 2009}}
{{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.

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.)

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.

In [[theory of relativity|relativistic]] [[quantum field theory]], a stable bound state of n particles with masses m1, ..., mn shows up as a [[pole (complex analysis)|pole]] in the [[S-matrix]] with a center of mass energy which is less than m1+...+mn. An [[unstable]] bound state (see [[resonance]]) shows up as a pole with a [[complex number|complex]] center of mass energy.

==Examples==
[[Image:Particle overview.svg|thumb|400px|An overview of the various families of elementary and composite particles, and the theories describing their interactions]]
* 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==
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:'''
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.

* 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.

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

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[[Category:Quantum mechanics]]
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