Advertisement

Inequalities pp 367-376 | Cite as

Quantum Coherent Operators: A Generalization of Coherent States

  • Elliott H. Lieb
  • Jan Philip Solovej

Abstract

We introduce a technique to compare different, but related, quantum systems, thereby generalizing the way that coherent states are used to compare quantum systems to classical systems in semiclassical analysis. We then use this technique to estimate the dependence of the free energy of the quantum Heisenberg model on the spin value, and to estimate the relation between the ferromagnetic and antiferromagnetic free energies.

Keywords

Coherent State Spin Operator Heisenberg Model National Science Foundation Grant Operator Completeness 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arecchi, F. T., Courtens, E., Gilmore, R., and Thomas, H., Atomic coherent states in quantum optics, Phys. Rev. A 6, 2211–2237 (1972).CrossRefGoogle Scholar
  2. 2.
    Bargmann, V., On a Hilbert space of analytic functions and an associated integral transform, part 1, Comm. Pure Appl. Math. 14, 187–214 (1961).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bargmann, V., On a Hilbert space of analytic functions and an associated integral transform, part 2, Comm. Pure Appl. Math. 20, 1–101 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berezin, F. A., Izv. Akad. Nauk SSSR Ser. Mat. 36(5), 1134–1167 (1972). English translation: Covariant and contravariant symbols of operators. Math. USSR-Izv. 6(5), 1117-1151 (1972) and F. A. Berezin. General concept of quantization, Comm. Math. Phys. 40, 153-174 (1975).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Feng, D. H., Gilmore, R., and Zhang, W-M., Coherent states: Theory and some applications, Rev. Mod. Phys. 62, 867–927 (1990).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Klauder, J. R., The action option and a Feynman quantization of spinor fields in terms of ordinary onumbers, Ann. Phys. 11, 123 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Klauder, J. R., and Skagerstam, B-S., Coherent States, World Scientific, Singapore, 1985.zbMATHGoogle Scholar
  8. 8.
    Lieb, E. H., The classical limit of quantum spin systems, Comm. Math. Phys. 31, 327–340 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Lieb, E. H., Proof of an entropy conjecture of Wehrl, Comm. Math. Phys. 62, 35–41 (1978).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Perelomov, A., Generalized Coherent States and their Applications, Springer-Verlag, New York, Berlin, Heidelberg, 1986.zbMATHCrossRefGoogle Scholar
  11. 11.
    Schrödinger, E., Der stetige übergang von der Mikro-zur Makromechanik, Naturwiss. 14, 664–666 (1926).zbMATHCrossRefGoogle Scholar
  12. 12.
    Segal, I. E., Mathematical characterizations of the physical vacuum for the linear Bose-Einstein fields, Illinois J. Math. 6, 500–523 (1962).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Simon, B., The classical limit of quantum partition functions, Comm. Math. Phys. 71, 247–276 (1980).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Thirring, W.E., A lower bound with the best possible constant for Coulomb hamiltonians, Comm. Math. Phys. 79, 1–7 (1981).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wehrl, A., On the relation between classical and quantum-mechanical entropy. Rep. Math. Phys. 12, 385 (1977).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott H. Lieb
    • 2
  • Jan Philip Solovej
    • 1
    • 3
  1. 1.Institute for Advanced StudySchool of MathematicsPrincetonUSA
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations