Bound state

Summary

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.[1]

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.[2] 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".[3] Examples include radionuclides and Rydberg atoms.[4]

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.

Examples

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

Definition

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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   on  . Let   be the induced probability distribution of   with respect to  . Then the evolution

 

is bound with respect to   if

 ,

where  .[dubious – discuss][9]

A quantum particle is in a bound state if at no point in time it is found “too far away" from any finite region  . 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 for all times  .[10] Furthermore, a bound state lies within the pure point part of the spectrum of   if and only if it is an eigenstate of  .[11]

More informally, "boundedness" results foremost from the choice of domain of definition and characteristics of the state rather than the observable.[nb 1] For a concrete example: let   and let   be the position operator. Given compactly supported   and  .

  • If the state evolution of   "moves this wave package to the right", e.g. if   for all  , then   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   "just moves   inside a bounded domain", then   is bound with respect to position.

Properties

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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.[12][13]

Position-bound states

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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. 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.[14] Hence, negative energy-states are bound if V vanishes at infinity.

Non-Degeneracy in One dimensional bound states

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1D 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 wavefunction in higher dimensions. Due to the property of non-degenerate states, one dimensional bound states can always be expressed as real wavefunctions.

Node theorem

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Node theorem states that n-th bound wavefunction ordered according to increasing energy has exactly n-1 nodes, ie. points   where  . Due to the form of Schrödinger's time independent equations, it is not possible for a physical wavefunction to have   since it corresponds to   solution.[15]

Requirements

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

 ,

where  , g is the gauge coupling constant, and ƛi = /mic 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 m1 and m2, the Bohr radius of the system becomes

 

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/c2, 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.[16]

See also

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Remarks

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References

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  1. ^ "Bound state - Oxford Reference".
  2. ^ Blanchard, Philippe; Brüning, Erwin (2015). Mathematical Methods in Physics. Birkhäuser. p. 430. ISBN 978-3-319-14044-5.
  3. ^ 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.
  4. ^ Gallagher, Thomas F. (1994-09-15). "Oscillator strengths and lifetimes". Rydberg Atoms (1 ed.). Cambridge University Press. pp. 38–49. doi:10.1017/cbo9780511524530.005. ISBN 978-0-521-38531-2.
  5. ^ 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.
  6. ^ 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.
  7. ^ 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.
  8. ^ Max T. C. Wong & C. K. Law (May 2011). "Two-polariton bound states in the Jaynes-Cummings-Hubbard model". Phys. Rev. A. 83 (5). American Physical Society: 055802. arXiv:1101.1366. Bibcode:2011PhRvA..83e5802W. doi:10.1103/PhysRevA.83.055802. S2CID 119200554.
  9. ^ Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: I: Functional analysis. Academic Press. p. 303. ISBN 978-0-12-585050-6.
  10. ^ Ruelle, D. (1969). "A remark on bound states in potential-scattering theory" (PDF). Il Nuovo Cimento A. 61 (4). Springer Science and Business Media LLC. doi:10.1007/bf02819607. ISSN 0369-3546.
  11. ^ Simon, B. (1978). "An Overview of Rigorous Scattering Theory". p. 3.
  12. ^ Stillinger, Frank H.; Herrick, David R. (1975). "Bound states in the continuum". Physical Review A. 11 (2). American Physical Society (APS): 446–454. doi:10.1103/physreva.11.446. ISSN 0556-2791.
  13. ^ Hsu, Chia Wei; Zhen, Bo; Stone, A. Douglas; Joannopoulos, John D.; Soljačić, Marin (2016). "Bound states in the continuum". Nature Reviews Materials. 1 (9). Springer Science and Business Media LLC. doi:10.1038/natrevmats.2016.48. hdl:1721.1/108400. ISSN 2058-8437.
  14. ^ Hall, Brian C. (2013). Quantum theory for mathematicians. Graduate texts in mathematics. New York Heidelberg$fDordrecht London: Springer. p. 316-320. ISBN 978-1-4614-7115-8.
  15. ^ Berezin, F. A. (1991). The Schrödinger equation. Dordrecht ; Boston : Kluwer Academic Publishers. pp. 64–66. ISBN 978-0-7923-1218-5.
  16. ^ 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.

Further reading

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  • 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.
  • Ruelle, David (9 January 2016). "A Remark on Bound States in Potential-Scattering Theory" (PDF). Nuovo Cimento A. 61 (June 1969): 655–662. doi:10.1007/BF02819607. S2CID 56050354. Retrieved 27 December 2021.