In physics, a bound state describes a system where a particle is subject to a 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 tunnelled through in order to decay. This is true for some 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 square-integrable wavefunctions exist (normalized in the range of the potential). The energies of these wavefunctions are negative.
In relativistic quantum field theory, a stable bound state of n particles with masses m1, … , mn shows up as a 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 center of mass energy.
- 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 states are unstable and will decay into bound states with less energy by emitting a photon.
- A nucleus is a bound state of protons and neutrons (nucleons).
- A positronium "atom" is an unstable bound state of an electron and a positron. It decays into photons.
- 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 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. 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.
In mathematical quantum physics
Let H be a complex separable Hilbert space, be a one-parametric group of unitary operators on H and be a statistical operator on H. Let A be an observable on H and let be the induced probability distribution of A with respect to ρ on the Borel σ-algebra on . Then the evolution of ρ induced by U is said to be bound with respect to A if , where .
Example: Let and let A be the position observable. Let have compact support and .
- If the state evolution of ρ "moves this wave package constantly to the right", e.g. if for all , then ρ is not a bound state with respect to the position.
- If does not change in time, i.e. for all , then is a bound state with respect to position.
- More generally: If the state evolution of ρ "just moves ρ inside a bounded domain", then ρ is also a bound state with respect to position.
It should be emphasized that a bound state can have its energy located in the continuum spectrum. This fact was first pointed out by John von Neumann and Eugene Wigner in 1929.  This exotic type of bound state has been realized in several simple models.  
- K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. H. Denschlag, A. J. Daley, A. Kantian, H. P. Buchler and P. Zoller (2006). "Repulsively bound atom pairs in an optical lattice". Nature 441: 853. arXiv:cond-mat/0605196. Bibcode:2006Natur.441..853W. doi:10.1038/nature04918.
- Javanainen, Juha and Odong, Otim and Sanders, Jerome C. (Apr 2010). "Dimer of two bosons in a one-dimensional optical lattice". Phys. Rev. A (American Physical Society) 81 (4): 043609. arXiv:1004.5118. Bibcode:2010PhRvA..81d3609J. doi:10.1103/PhysRevA.81.043609.
- M. Valiente and D. Petrosyan (2008). "Two-particle states in the Hubbard model". J. Phys. B: At. Mol. Opt. Phys. 41: 161002. Bibcode:2008JPhB...41p1002V. doi:10.1088/0953-4075/41/16/161002.
- Max T. C. Wong and 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.
- von Neumann, John; Wigner, Eugene (1929). "Über merkwürdige diskrete Eigenwerte". Physikalische Zeitschrift 30: 465–467.
- Zhang, J M; Braak, Daniel; Kollar, Marcus (2012). "Bound States in the Continuum Realized in the One-Dimensional Two-Particle Hubbard Model with an Impurity". Physical Review Letters 109: 116405. arXiv:1205.6431. Bibcode:2012PhRvL.109k6405Z. doi:10.1103/PhysRevLett.109.116405.
- Longhi, S; Della Valle, G (2013). "Tamm-Hubbard surface states in the continuum". J. Phys.: Condens. Matter. 235601: 235601. arXiv:1306.0658. Bibcode:2013JPCM...25w5601L. doi:10.1088/0953-8984/25/23/235601.