Bound state

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In quantum physics, a bound state is a special quantum state of a particle 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 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.


An overview of the various families of elementary and composite particles, and the theories describing their interactions


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 ][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]

Position-bound states[edit]

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

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

See also[edit]


  1. ^ Sakurai, Jun (1995). "7.8". In Tuan, San. 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.  (Formatting in original.)
  2. ^ 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: 853. arXiv:cond-mat/0605196Freely accessible. Bibcode:2006Natur.441..853W. doi:10.1038/nature04918. PMID 16778884. 
  3. ^ Javanainen, Juha; Odong Otim; 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:1004.5118Freely accessible. Bibcode:2010PhRvA..81d3609J. doi:10.1103/PhysRevA.81.043609. 
  4. ^ M. Valiente & 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. 
  5. ^ 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:1101.1366Freely accessible. Bibcode:2011PhRvA..83e5802W. doi:10.1103/PhysRevA.83.055802. 
  6. ^ von Neumann, John; Wigner, Eugene (1929). "Über merkwürdige diskrete Eigenwerte". Physikalische Zeitschrift. 30: 465–467.