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In quantum physics, a **bound state** is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. 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 vanishing at infinity, negative-energy states must be bound. In general, the energy spectrum of the set of bound states is discrete, unlike 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".^{[1]} Examples include certain radionuclides and electrets.^{[clarification needed]}^{[citation needed]}

In relativistic quantum field theory, a stable bound state of n particles with masses corresponds to a pole in the S-matrix with a center-of-mass energy less than . An unstable bound state shows up as a pole with a complex center-of-mass energy.

- 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 – namely the hydrogen atom – is formed. Only the lowest-energy bound state, the ground state, is stable. Other excited states 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 unstable bound state of an electron and a positron. It decays into photons.
- Any state in the quantum harmonic oscillator is bound, but has positive energy. Note that , so the below does not apply.
- A nucleus is a bound state of protons and neutrons (nucleons).
- The proton itself is a bound state of three quarks (two up and one down; one red, one green and one blue). However, unlike the case of the hydrogen atom, the individual quarks can never be isolated. See confinement.
- The Hubbard and 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.
^{[2]}^{[3]}^{[4]}The JCH Hamiltonian also supports two-polariton bound states when the photon-atom interaction is sufficiently strong.^{[5]}

Let H be a complex separable Hilbert space, be a one-parameter group of unitary operators on H and be a statistical operator on H. Let A be an observable on H and be the induced probability distribution of A with respect to ρ on the Borel σ-algebra of . Then the evolution of ρ induced by U is **bound** with respect to A if , where .^{[dubious – discuss]}^{[citation needed]}

More informally, a bound state is contained within a bounded portion of the spectrum of A. For a concrete example: let and let A be position. Given compactly-supported and .

- If the state evolution of ρ "moves this wave package constantly to the right", e.g. if for all , then ρ is not bound state with respect to position.
- If does not change in time, i.e. for all , then is bound with respect to position.
- More generally: If the state evolution of ρ "just moves ρ inside a bounded domain", then ρ is bound with respect to position.

Let A have measure-space codomain . A quantum particle is in a bound state if it is never found “too far away from any finite region ”, i.e. using a wavefunction representation,

Consequently, is finite. In other words, a state is a bound state if and only if it is finitely normalizable.

As finitely normalizable states must lie within the discrete part of the spectrum, bound states must lie within the discrete part. However, as Neumann and Wigner pointed out, a bound state can have its energy located in the continuum spectrum.^{[6]} In that case, bound states still are part of the discrete portion of the spectrum, but appear as Dirac masses in the spectral measure.^{[citation needed]}

Consider the one-particle Schrödinger equation. If a state has energy , then the wavefunction ψ satisfies, for some

so that ψ is exponentially suppressed at large x.^{[dubious – discuss]} Hence, negative energy-states are bound if V vanishes at infinity.

A boson with mass *m _{χ}* mediating a weakly coupled interaction produces an Yukawa-like interaction potential,

- ,

where , *g* is the gauge coupling constant, and *ƛ _{i}* = ℏ/

and yields the dimensionless number

- .

In order for the first bound state to exist at all, . Because the photon is massless, *D* is infinite for electromagnetism. For the weak interaction, the Z boson's mass is 91.1876±0.0021 GeV/*c*^{2}, which prevents the formation of bound states between most particles, as it is 97.2 times the proton's mass and 178,000 times the electron's mass.

Note however that if the Higgs interaction didn't break electroweak symmetry at the electroweak scale, then the SU(2) weak interaction would become confining.^{[7]}

**^**Sakurai, Jun (1995). "7.8". In Tuan, San (ed.).*Modern Quantum Mechanics*(Revised ed.). Reading, Mass: Addison-Wesley. pp. 418–9. ISBN 0-201-53929-2.Suppose the barrier were infinitely high ... we expect bound states, with energy

*E*> 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.**^**K. Winkler; G. Thalhammer; F. Lang; R. Grimm; J. H. Denschlag; A. J. Daley; A. Kantian; H. P. Buchler; P. Zoller (2006). "Repulsively bound atom pairs in an optical lattice".*Nature*.**441**(7095): 853–856. arXiv:cond-mat/0605196. Bibcode:2006Natur.441..853W. doi:10.1038/nature04918. PMID 16778884. S2CID 2214243.**^**Javanainen, Juha; Odong Otim; Sanders, Jerome C. (Apr 2010). "Dimer of two bosons in a one-dimensional optical lattice".*Phys. Rev. A*.**81**(4): 043609. arXiv:1004.5118. Bibcode:2010PhRvA..81d3609J. doi:10.1103/PhysRevA.81.043609. S2CID 55445588.**^**M. Valiente & D. Petrosyan (2008). "Two-particle states in the Hubbard model".*J. Phys. B: At. Mol. Opt. Phys*.**41**(16): 161002. arXiv:0805.1812. Bibcode:2008JPhB...41p1002V. doi:10.1088/0953-4075/41/16/161002. S2CID 115168045.**^**Max T. C. Wong & C. K. Law (May 2011). "Two-polariton bound states in the Jaynes-Cummings-Hubbard model".*Phys. Rev. A*. American Physical Society.**83**(5): 055802. arXiv:1101.1366. Bibcode:2011PhRvA..83e5802W. doi:10.1103/PhysRevA.83.055802. S2CID 119200554.**^**von Neumann, John; Wigner, Eugene (1929). "Über merkwürdige diskrete Eigenwerte".*Physikalische Zeitschrift*.**30**: 465–467.**^**Claudson, M.; Farhi, E.; Jaffe, R. L. (1 August 1986). "Strongly coupled standard model".*Physical Review D*.**34**(3): 873–887. Bibcode:1986PhRvD..34..873C. doi:10.1103/PhysRevD.34.873. PMID 9957220.

- Blanchard, Philippe; Brüning, Edward (2015). "Some Applications of the Spectral Representation".
*Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics*(2nd ed.). Switzerland: Springer International Publishing. p. 431. ISBN 978-3-319-14044-5.

- Gustafson, Stephen J.; Sigal, Israel Michael (2011). "Spectrum and Dynamics".
*Mathematical Concepts of Quantum Mechanics*(2nd ed.). Berlin, Heidelberg: Springer-Verlag. p. 50. ISBN 978-3-642-21865-1.