# Bound state

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 Template:Mvar particles with masses Template:Bigmath shows up as a pole in the S-matrix with a center of mass energy which is less than Template:Bigmath. An unstable bound state (see resonance) shows up as a pole with a complex center of mass energy.

## Examples

- 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.
^{[1]}^{[2]}^{[3]}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.^{[4]}

## In mathematical quantum physics

Let Template:Mvar be a complex separable Hilbert space, be a one-parametric group of unitary operators on Template:Mvar and be a statistical operator on Template:Mvar. Let Template:Mvar be an observable on Template:Mvar and let be the induced probability distribution of Template:Mvar with respect to Template:Mvar on the Borel σ-algebra on . Then the evolution of Template:Mvar induced by Template:Mvar is said to be **bound** with respect to Template:Mvar if , where .

**Example:**
Let and let Template:Mvar be the position observable. Let have compact support and .

- If the state evolution of Template:Mvar "moves this wave package constantly to the right", e.g. if for all , then Template:Mvar is not a bound state with respect to the position.

- More generally: If the state evolution of Template:Mvar "just moves Template:Mvar inside a bounded domain", then Template:Mvar 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. ^{[5]} This exotic type of bound state has been realized in several simple models. ^{[6]} ^{[7]}

## See also

## References

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