Descriptions of operators in quantum mechanics

  • N Mukunda
Article

Abstract

The problem of expressing a general dynamical variable in quantum mechanics as a function of a primitive set of operators is studied from several points of view. In the context of the Heisenberg commutation relation, the Weyl representation for operators and a new Fourier-Mellin representation are related to the Heisenberg group and the groupSL(2,R) respectively. The description of unitary transformations via generating functions is analysed in detail. The relation between functions and ordered functions of noncommuting operators is discussed, and results closely paralleling classical results are obtained.

Keywords

Heisenberg group Weyl representation Fourier-Mellin representation for operators functions of operators 

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References

  1. [1]
    Agarwal G S and Wolf E 1970Phys. Rev. D2, 2161CrossRefMathSciNetGoogle Scholar
  2. [2]
    Bargmann V 1947Ann. Math. 48 568CrossRefMathSciNetGoogle Scholar
  3. [3]
    Biedenharn L C and Louck J D 1971Ann. Phys. (NY) 63 459CrossRefMathSciNetGoogle Scholar
  4. [4]
    Cahill K and Glauber R J 1969Phys. Rev. 177 1857CrossRefGoogle Scholar
  5. [5]
    Caratheodory C 1965Calculus of variations and partial differential equations of the first order Part I (San Francisco, California: Holden-Day)MATHGoogle Scholar
  6. [6]
    Dirac P A M 1933Phys. Z. Sowjetunion 3 64MATHGoogle Scholar
  7. [7]
    Dirac P A M 1945Rev. Mod. Phys. 17 195CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    Dirac P A M 1958The principles of quantum mechanics 4th ed (Oxford: Clarendon Press)MATHGoogle Scholar
  9. [9]
    Edmonds A R 1957Angular momentum in quantum mechanics (Princeton, N. J.: Princeton University Press)MATHGoogle Scholar
  10. [10]
    Eisenhart L P 1933Continuous groups of transformations (New York: Dover)MATHGoogle Scholar
  11. [11]
    Fano U and Racah G 1959Irreducible tensorial sets (New York: Academic Press)Google Scholar
  12. [12]
    Gel’fand I M, Graev M I and Vilenkin N Ya 1966Generalised functions (New York: Academic Press) Vol. 5Google Scholar
  13. [13]
    Goshen S and Lipkin H J 1959Ann. Phys. (NY) 6 301MathSciNetMATHGoogle Scholar
  14. [14]
    Jordan P 1926Z. Phys. 38 513CrossRefGoogle Scholar
  15. [15]
    Jordan T F 1969Linear operators for quantum mechanics (New York: John Wiley)MATHGoogle Scholar
  16. [16]
    Klauder J R and Sudarshan E C G 1968Fundamentals of quantum optics (New York: W A Benjamin)Google Scholar
  17. [17]
    Mukunda N 1974Pramana 2 1CrossRefGoogle Scholar
  18. [18]
    Pool J C T 1966J. Math. Phys. 7 66CrossRefMathSciNetMATHGoogle Scholar
  19. [19]
    Saletan E J and Cromer A H 1971Theoretical mechanics (New York: John Wiley)MATHGoogle Scholar
  20. [20]
    de Shalit A and Talmi I 1963Nuclear shell theory (New York and London: Academic Press)Google Scholar
  21. [21]
    Weyl H 1931The theory of Groups and quantum mechanics (New York: Dover)MATHGoogle Scholar
  22. [22]
    Whittaker E T 1927A treatise on the analytical dynamics of particles and rigid bodies’ 3rd ed. (Cambridge: University Press)Google Scholar

Copyright information

© Indian Academy of Sciences 1979

Authors and Affiliations

  • N Mukunda
    • 1
  1. 1.Centre for Theoretical StudiesIndian Institute of ScienceBangalore

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