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.

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. The energy spectrum of the set of bound states are most commonly discrete, unlike scattering states of 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". Examples include radionuclides and Rydberg atoms.

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

Examples

• 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 , 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. The JCH Hamiltonian also supports two-polariton bound states when the photon-atom interaction is sufficiently strong.

Definition

Let σ-finite measure space be a probability space associated with separable complex Hilbert space. Define a one-parameter group of unitary operators, a density operator and an observable. Then the evolution

is bound if

1. ,

.

A quantum particle is in a bound state. Using a wave function representation, for example, this means

such that

In general, a quantum state is a bound state if and only if it is finitely normalizable and remains spatially localized. Furthermore, a bound state lies within the pure point part if and only if it is an eigenvector.

More informally, "boundedness" results foremost from the choice of domain of definition and characteristics of the state rather than the observable. be the position operator.

• is not bound state with respect to position.

• is bound with respect to position.

• is bound with respect to position.

Properties

As finitely normalizable states must lie within the pure point part of the spectrum, bound states must lie within the pure point part. However, as 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.

Position-bound states

, then the wavefunction

so that is exponentially suppressed at large . 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. 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.

Node theorem

solution.

Requirements

A boson with mass mediating a weakly coupled interaction produces an Yukawa-like interaction potential,

1. ,

, is the gauge coupling constant, and 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 and , the Bohr radius of the system becomes

and yields the dimensionless number

1. .

. Because the photon is massless, is infinite for electromagnetism. For the weak interaction, the Z boson's mass is , which prevents the formation of bound states between most particles, as it is the proton's mass and the electron's mass.

Note, however, that, if the Higgs interaction did not break electroweak symmetry at the electroweak scale, then the SU(2) weak interaction would become confining.

See also

• Bethe–Salpeter equation

• Bound state in the continuum

• Composite field

• Cooper pair

• Exciton

• Resonance (particle physics)

• Levinson's theorem

Remarks

References

Further reading