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{{Short description|Quantum physics terminology}}

A '''bound state''' is a composite of two or more fundamental building blocks, such as particles, atoms, or bodies, that behaves as a single object and in which energy is required to split them.<ref>{{Cite web|url=https://www.oxfordreference.com/display/10.1093/oi/authority.20110803095520865|title=Bound state - Oxford Reference}}</ref>

In [[quantum physics]], a bound state is a [[quantum state]] of a [[particle]] subject to a [[potential energy|potential]] such that the particle has a tendency to remain localized in one or more regions of space.<ref>{{cite book | last=Blanchard | first=Philippe | last2=Brüning | first2=Erwin | title=Mathematical Methods in Physics | publisher=Birkhäuser | date=2015 | isbn=978-3-319-14044-5|page=430}}</ref> The potential may be external or it may be the result of the presence of another particle; in the latter case, one can equivalently define a bound state as a state representing two or more particles whose [[interaction energy]] exceeds the total energy of each separate particle. One consequence is that, given a potential [[vanish at infinity|vanishing at infinity]], negative-energy states must be bound.  The [[energy spectrum]] of the set of bound states are most commonly discrete, unlike [[scattering state]]s of [[Free particle|free particles]], which have a continuous spectrum.

Although not bound states in the strict sense, metastable states with a net positive interaction energy, but long decay time, are often considered unstable bound states as well and are called "quasi-bound states".<ref>{{cite book |last1=Sakurai |first1=Jun |editor1-last=Tuan |editor1-first=San |title=Modern Quantum Mechanics |date=1995 |publisher=Addison-Wesley |location=Reading, Mass |isbn=0-201-53929-2 |pages=418–9 |edition=Revised |chapter=7.8 |quote=Suppose the barrier were infinitely high ... we expect bound states, with energy ''E''&nbsp;>&nbsp;0. ... They are ''stationary'' states with infinite lifetime.  In the more realistic case of a finite barrier, the particle can be trapped inside, but it cannot be trapped forever.  Such a trapped state has a finite lifetime due to quantum-mechanical tunneling. ... Let us call such a state '''quasi-bound state''' because it would be an honest bound state if the barrier were infinitely high.}}</ref>  Examples include [[radionuclides]] and [[Rydberg atom]]s.<ref>{{Cite book |last=Gallagher |first=Thomas F. |url=https://www.cambridge.org/core/product/identifier/9780511524530/type/book |title=Rydberg Atoms |date=1994-09-15 |publisher=Cambridge University Press |isbn=978-0-521-38531-2 |edition=1 |pages=38–49 |chapter=Oscillator strengths and lifetimes |doi=10.1017/cbo9780511524530.005}}</ref>

In [[special relativity|relativistic]] [[quantum field theory]], a stable bound state of {{mvar|n}} particles with masses <math>\{m_k\}_{k=1}^n</math> corresponds to a [[pole (complex analysis)|pole]] in the [[S-matrix]] with a [[center of mass frame|center-of-mass energy]] less than <math>\textstyle\sum_k m_k</math>.  An [[unstable]] bound state 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; when they do, 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&nbsp;– namely the [[hydrogen atom]]&nbsp;– is formed. Only the lowest-energy bound state, the [[ground state]], is stable. Other [[excited state]]s are unstable and will decay into stable (but not other unstable) bound states with less energy by emitting a [[photon]].
*A [[positronium]] "atom" is an [[resonance|unstable bound state]] of an [[electron]] and a [[positron]]. It decays into [[photon]]s.
*Any state in the [[quantum harmonic oscillator]] is bound, but has positive energy.  Note that <math>\lim_{x\to\pm\infty}{V_{\text{QHO}}(x)} = \infty</math> , so the [[#Normalization|below]] <!-- Dead link --> does not apply.
*A [[atomic nucleus|nucleus]] is a bound state of [[proton]]s and [[neutron]]s ([[nucleon]]s).
*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]].
*The [[Hubbard model|Hubbard]] and [[Jaynes-Cummings-Hubbard model|Jaynes–Cummings–Hubbard (JCH)]] models support similar bound states.  In the Hubbard model, two repulsive [[bosonic]] [[atoms]] can form a bound pair in an [[optical lattice]].<ref>
{{cite journal
|author1=K. Winkler |author2=G. Thalhammer |author3=F. Lang |author4=R. Grimm |author5=J. H. Denschlag |author6=A. J. Daley |author7=A. Kantian |author8=H. P. Buchler |author9=P. Zoller | title = Repulsively bound atom pairs in an optical lattice
|journal = [[Nature (journal)|Nature]]
| year = 2006
| volume = 441
|issue=7095 | pages = 853–856
|arxiv = cond-mat/0605196 |bibcode = 2006Natur.441..853W |doi = 10.1038/nature04918 | pmid=16778884|s2cid=2214243 }}
</ref><ref>
{{cite journal
| title = Dimer of two bosons in a one-dimensional optical lattice
|author1=Javanainen, Juha |author2=Odong Otim |author3=Sanders, Jerome C. | journal = [[Phys. Rev. A]]
| volume = 81
| issue = 4
| pages = 043609
|date=Apr 2010
| doi = 10.1103/PhysRevA.81.043609
|arxiv = 1004.5118 |bibcode = 2010PhRvA..81d3609J |s2cid=55445588 }}
</ref><ref>
{{cite journal
|author1=M. Valiente  |author2=D. Petrosyan
 |name-list-style=amp | title = Two-particle states in the Hubbard model
| journal = J. Phys. B: At. Mol. Opt. Phys.
| year = 2008
| volume = 41
|issue=16
 | pages = 161002
| doi=10.1088/0953-4075/41/16/161002
|bibcode = 2008JPhB...41p1002V |arxiv=0805.1812|s2cid=115168045
 }}
</ref> The JCH Hamiltonian also supports two-[[polariton]] bound states when the photon-atom interaction is sufficiently strong.<ref>
{{cite journal
| title = Two-polariton bound states in the Jaynes-Cummings-Hubbard model
|author1=Max T. C. Wong  |author2=C. K. Law
 |name-list-style=amp | journal = [[Phys. Rev. A]]
| volume = 83
| issue = 5
| pages = 055802
|date=May 2011
| doi = 10.1103/PhysRevA.83.055802
| publisher = [[American Physical Society]]
|arxiv = 1101.1366 |bibcode = 2011PhRvA..83e5802W |s2cid=119200554 }}
</ref>

==Definition==
{{see also|Decomposition of spectrum (functional analysis) #Quantum mechanics}}
Let [[Measure space#Important classes of measure spaces|{{math|''&sigma;''}}-finite measure space]] <math>(X, \mathcal A, \mu)</math> be a [[Measure space#Important classes of measure spaces|probability space]] associated with [[Separable space|separable]] [[complex number|complex]] [[Hilbert space]] <math>H</math>. Define a [[Stone's theorem on one-parameter unitary groups|one-parameter group of unitary operators]] <math> (U_t)_{t\in \mathbb{R}} </math>, a [[density operator]] <math>\rho = \rho(t_0) </math> and an [[observable]] <math>T</math> on <math>H</math>. Let <math>\mu(T,\rho)</math> be the induced probability distribution of <math>T</math> with respect to <math>\rho</math>. Then the evolution
:<math>\rho(t_0)\mapsto [U_t(\rho)](t_0) = \rho(t_0 +t)</math> 
is '''bound''' with respect to <math>T</math> if 
:<math>\lim_{R \rightarrow \infty}{\sup_{t \geq t_0}{\mu(T,\rho(t))(\mathbb{R}_{> R})}} = 0 </math>, 
where <math>\mathbb{R}_{>R} = \lbrace x \in \mathbb{R} \mid x > R \rbrace </math>.{{dubious|date=November 2016}}<ref>{{cite book | last1=Reed | first1=M. | last2=Simon | first2=B. | title=Methods of Modern Mathematical Physics: I: Functional analysis | publisher=Academic Press | year=1980 |page=303 | isbn=978-0-12-585050-6}}</ref>

A quantum particle is in a '''bound state''' if at no point in time it is found “too far away" from any finite region <math>R\subset X</math>. Using a [[wave function]] representation, for example, this means<ref>{{cite book | last=Gustafson | first=Stephen J. | last2=Sigal | first2=Israel Michael | title=Mathematical Concepts of Quantum Mechanics | publisher=Springer International Publishing | publication-place=Cham | year=2020 | isbn=978-3-030-59561-6 | issn=0172-5939 | doi=10.1007/978-3-030-59562-3|chapter=Bound and Decaying States}}</ref>
:<math>\begin{align}
0 &= \lim_{R\to\infty}{\mathbb{P}(\text{particle measured inside }X\setminus R)} \\
&= \lim_{R\to\infty}{\int_{X\setminus R}|\psi(x)|^2\,d\mu(x)},
\end{align}</math>
such that
:<math>\int_X{|\psi(x)|^{2}\,d\mu(x)} < \infty.</math>
In general, a quantum state is a bound state ''if and only if'' it is finitely [[Probability amplitude#Normalization|normalizable]] for all times <math>t\in\mathbb{R}</math> and remains spatially localized.<ref>{{cite journal | last=Ruelle | first=D. | title=A remark on bound states in potential-scattering theory | journal=Il Nuovo Cimento A | publisher=Springer Science and Business Media LLC | volume=61 | issue=4 | year=1969 | issn=0369-3546 | doi=10.1007/bf02819607 | url=https://www.ihes.fr/%7Eruelle/PUBLICATIONS/%5B25%5D.pdf}}</ref> Furthermore, a bound state lies within the [[Spectrum_(functional_analysis)#Classification_of_points_in_the_spectrum|pure point part]] of the spectrum of <math>T</math> ''if and only if'' it is an [[eigenvector]] of <math>T</math>.<ref>{{cite web | last=Simon | first=B. | title=An Overview of Rigorous Scattering Theory | date=1978 |page=3| url=https://api.semanticscholar.org/CorpusID:16913591}}</ref> 

More informally, "boundedness" results foremost from the choice of [[domain of definition]] and characteristics of the state rather than the observable.<ref group=nb>See [[Expectation_value_(quantum_mechanics)#Example_in_configuration_space|Expectation value (quantum mechanics)]] for an example.</ref> For a concrete example: let <math>H := L^2(\mathbb{R}) </math> and let <math>T</math> be the [[position operator]]. Given compactly supported <math>\rho = \rho(0) \in H</math> and <math>[-1,1] \subseteq \mathrm{Supp}(\rho)</math>.

*If the state evolution of <math>\rho</math> "moves this wave package 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 bound state with respect to 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 bound 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 bound with respect to position.

==Properties==
{{See also|Spectrum (physical sciences) #Continuous versus discrete spectra}}
As finitely normalizable states must lie within the [[Spectrum_(functional_analysis)#Classification_of_points_in_the_spectrum|pure point part]] of the spectrum, bound states must lie within the pure point part. However, as [[John von Neumann|Neumann]] and [[Wigner]] pointed out, it is possible for the energy of a bound state to be located in the continuous part of the spectrum. This phenomenon is referred to as [[bound state in the continuum]].<ref>{{cite journal | last1=Stillinger | first1=Frank H. | last2=Herrick | first2=David R. | title=Bound states in the continuum | journal=Physical Review A | publisher=American Physical Society (APS) | volume=11 | issue=2 | year=1975| issn=0556-2791 | doi=10.1103/physreva.11.446 | pages=446–454}}</ref><ref>{{cite journal | last1=Hsu | first1=Chia Wei | last2=Zhen | first2=Bo | last3=Stone | first3=A. Douglas | last4=Joannopoulos | first4=John D. | last5=Soljačić | first5=Marin | title=Bound states in the continuum | journal=Nature Reviews Materials | publisher=Springer Science and Business Media LLC | volume=1 | issue=9 | date=2016 | url=https://dspace.mit.edu/handle/1721.1/108400 | issn=2058-8437 | doi=10.1038/natrevmats.2016.48| hdl=1721.1/108400 | hdl-access=free }}</ref>

===Position-bound states===
Consider the one-particle Schrödinger equation. If a state has energy <math display="inline"> E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}</math>, then the wavefunction {{mvar|ψ}} satisfies, for some <math>X > 0</math>

:<math>\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X</math>

so that {{mvar|ψ}} is exponentially suppressed at large {{mvar|x}}. This behaviour is well-studied for smoothly varying potentials in the [[WKB approximation]] for wavefunction, where an oscillatory behaviour is observed if the right hand side of the equation is negative and growing/decaying behaviour if it is positive.<ref>{{Cite book |last=Hall |first=Brian C. |title=Quantum theory for mathematicians |date=2013 |publisher=Springer |isbn=978-1-4614-7115-8 |series=Graduate texts in mathematics |location=New York Heidelberg$fDordrecht London |page=316-320}}</ref> Hence, negative energy-states are bound if <math>V(x)</math> vanishes at infinity. 

=== Non-degeneracy in one-dimensional bound states ===
One-dimensional bound states can be shown to be non-degenerate in energy for well-behaved wavefunctions that decay to zero at infinities. This need not hold true for wavefunctions in higher dimensions. Due to the property of non-degenerate states, one-dimensional bound states can always be expressed as real wavefunctions.

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Proof
|-
|
Consider two energy eigenstates states <math display="inline"> \Psi_1</math> and <math display="inline"> \Psi_2</math> with same energy eigenvalue.

Then since, the Schrodinger equation, which is expressed as:<math display="block">E  = - \frac 1 {\Psi_i(x,t)} \frac{\hbar^2}{2m}\frac{\partial^2\Psi_i(x,t) }{\partial x^2} + V(x,t) </math>is satisfied for i = 1 and 2, subtracting the two equations gives:<math display="block">\frac 1 {\Psi_1(x,t)} \frac{\partial^2\Psi_1(x,t) }{\partial x^2} - \frac 1 {\Psi_2(x,t)} \frac{\partial^2\Psi_2(x,t) }{\partial x^2} = 0     </math>which can be rearranged to give the condition:<math display="block">
\frac{\partial }{\partial x} \left(\frac{\partial \Psi_1}{\partial x}\Psi_2\right)-\frac{\partial }{\partial x} \left(\frac{\partial \Psi_2}{\partial x}\Psi_1\right)=0  </math>Since <math display="inline">
\frac{\partial \Psi_1}{\partial x}(x)\Psi_2(x)- 
\frac{\partial \Psi_2}{\partial x}(x)\Psi_1(x)= C  </math>, taking limit of x going to infinity on both sides, the wavefunctions vanish and gives <math display="inline">
C = 0  </math>.


Solving for <math display="inline">
\frac{\partial \Psi_1}{\partial x}(x)\Psi_2(x) =
\frac{\partial \Psi_2}{\partial x}(x)\Psi_1(x) </math>, we get: <math display="inline">
\Psi_1(x) = k \Psi_2(x)       </math> which proves that the energy eigenfunction of a 1D bound state is unique.



Furthermore it can be shown that these wavefunctions can always be represented by a completely real wavefunction. Define real functions <math display="inline">
\rho_1(x) </math> and <math display="inline">
\rho_2(x)   </math> such that <math display="inline">
\Psi(x) = \rho_1(x) + i \rho_2(x) </math>. Then, from Schrodinger's equation:<math display="block">\Psi'' = - \frac{2m(E-V(x))}{\hbar^2}\Psi </math> we get that, since the terms in the equation are all real values:<math display="block">\rho_i'' = - \frac{2m(E-V(x))}{\hbar^2}\rho_i   </math>applies for i = 1 and 2. Thus every 1D bound state can be represented by completely real eigenfunctions. Note that real function representation of wavefunctions from this proof applies for all non-degenerate states in general. 
|}

=== Node theorem ===
Node theorem states that <math>n\text{th}</math> bound wavefunction ordered according to increasing energy has exactly <math>n-1</math> nodes, i.e., points <math>x=a</math> where <math>\psi(a)=0 \neq \psi'(a)</math>. Due to the form of Schrödinger's time independent equations, it is not possible for a physical wavefunction to have <math>\psi(a) = 0 = \psi'(a)</math> since it corresponds to <math>\psi(x)=0</math> solution.<ref>{{Cite book |last=Berezin |first=F. A. |url=http://archive.org/details/schrodingerequat0000bere |title=The Schrödinger equation |publisher=Dordrecht ; Boston : Kluwer Academic Publishers |year=1991 |isbn=978-0-7923-1218-5 |pages=64–66}}</ref>

==Requirements==

A [[boson]] with mass {{math|''m<sub>χ</sub>''}} [[gauge boson|mediating]] a [[Coupling (physics)#Particle physics and quantum field theory|weakly coupled interaction]] produces an [[Yukawa potential|Yukawa-like]] interaction potential,

:<math>V(r) = \pm\frac{\alpha_\chi}{r} e^{- \frac{r}{\lambda\!\!\!\frac{}{\ }_\chi}}</math>,<!--\bar\lambda is an overbar, \sout\lambda is missing-->

where <math>\alpha_\chi=g^2/4\pi</math>, {{math|''g''}} is the gauge coupling constant, and {{math|''ƛ<sub>i</sub>'' {{=}} {{sfrac|ℏ|''m<sub>i</sub>c''}}}} is the [[reduced Compton wavelength]]. A [[scalar boson]] produces a universally attractive potential, whereas a vector attracts particles to antiparticles but repels like pairs. For two particles of mass {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}}, the [[Bohr radius]] of the system becomes

:<math>a_0=\frac{{\lambda\!\!\!^{{}^\underline{\ \ }}}_1 + {\lambda\!\!\!^{{}^\underline{\ \ }}}_2}{\alpha_\chi}</math> <!--<span style="font-size: 1.25em;">{{math|''a''<sub>0</sub> {{=}} ''{{sfrac|ƛ<sub>1</sub> + ƛ<sub>2</sub>|α<sub>χ</sub>}}''}}</span>-->

and yields the dimensionless number

:<math>D=\frac{{\lambda\!\!\!^{{}^\underline{\ \ }}}_\chi}{a_0} = \alpha_\chi\frac{{\lambda\!\!\!^{{}^\underline{\ \ }}}_\chi}{{\lambda\!\!\!^{{}^\underline{\ \ }}}_1 + {\lambda\!\!\!^{{}^\underline{\ \ }}}_2} = \alpha_\chi\frac{m_1+m_2}{m_\chi}</math>.

In order for the first bound state to exist at all, <math>D\gtrsim 0.8</math>.  Because the [[photon]] is massless, {{math|''D''}} is infinite for [[electromagnetism]].  For the [[weak interaction]], the [[Z boson]]'s mass is {{val|91.1876|0.0021|u=GeV/c2}}, which prevents the formation of bound states between most particles, as it is {{val|97.2|u=times}} the [[proton]]'s mass and {{val|178,000|u=times|fmt=commas}} the [[electron]]'s mass.

Note, however, that, if the [[Higgs mechanism|Higgs interaction]] did not break electroweak symmetry at the [[electroweak scale]], then the SU(2) [[weak interaction]] would become [[Confinement (physics)|confining]].<ref>{{cite journal |last1=Claudson |first1=M. |last2=Farhi |first2=E. |last3=Jaffe |first3=R. L. |title=Strongly coupled standard model |journal=Physical Review D |date=1 August 1986 |volume=34 |issue=3 |pages=873–887 |doi=10.1103/PhysRevD.34.873 |pmid=9957220 |bibcode=1986PhRvD..34..873C }}</ref>

==See also==
*[[Bethe–Salpeter equation]]
*[[Bound state in the continuum]]
*[[Composite field]]
*[[Cooper pair]]
*[[Exciton]]
*[[Resonance (particle physics)]]
*[[Levinson's theorem]]

==Remarks==
{{Reflist|group=nb}}

==References==
{{Reflist|2}}

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{{DEFAULTSORT:Bound State}}
[[Category:Quantum field theory]]
[[Category:Quantum states]]

==Further reading==
* {{cite book |last1=Blanchard |first1=Philippe |last2=Brüning |first2=Edward |title=Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics |date=2015 |publisher=Springer International Publishing |location=Switzerland |isbn=978-3-319-14044-5 |page=431 |edition=2nd |language=English |chapter=Some Applications of the Spectral Representation}}