The hypervirial and viral theorems in terms of Weyl symbols

  • L. Cohen


We formulate the hypervirial and virial theorems in terms of symbols. We use the Weyl correspondence to relate symbols to operators and express the standard operator forms of the hyperverial and virial theorems in terms of Weyl symbols and Wigner distribution. We obtain explicit expressions for the relevant symbols. We also give a phase space operator formulation. Special cases are considered, and the theorem of Molahajloo is shown to be a such a case. We also consider the off-diagonal hyperverial theorem.


Hypervirial theorem Virial theorem Weyl symbols Wigner distribution Generalized phase-space distributions 

Mathematics Subject Classification

Primary 47G30 Secondary 81S30 


  1. 1.
    Bader, F.W.: Atoms in Molecules—A Quantum Theory. Oxford University Press, Oxford (1990)Google Scholar
  2. 2.
    Chen, J.C.Y.: Off-diagonal hypervirial theorem and its applications. J. Chem. Phys. 40, 615 (1964)CrossRefGoogle Scholar
  3. 3.
    Cohen, L.: Generalized phase space distribution functions. J. Math. Phys. 7, 781–786 (1966)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cohen, L.: Local kinetic energy in quantum mechanics. J. Chem. Phys. 70, 788 (1979)CrossRefGoogle Scholar
  5. 5.
    Cohen, L.: Local virial and tensor theorems. J. Phys. Chem. A 115(12919), 2011 (2011)Google Scholar
  6. 6.
    Cohen, L.: The eigenvalue problem in phase space. J. Comput. Chem. 38, 2453–2552 (2017)CrossRefGoogle Scholar
  7. 7.
    Cohen, L.: The Weyl Operator and Its Generalization. Springer, Basel (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Epstein, S.: The Variation Method in Quantum Chemistry. Academic Press, New York (1974)Google Scholar
  9. 9.
    Fernandez, F.M., Castro, E.A.: Hypervirial Theorems. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  10. 10.
    Galleani, L., Cohen, L.: The Wigner distribution for classical systems. Phys. Lett. A 302, 149–155 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hirschfelder, J.O.: Classical and quantum mechanical hypervirial theorems. J. Chem. Phys. 33, 1462 (1960)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lee, H.W.: Theory and application of quantum phase-space distribution functions. Phys. Rep. 259, 147–211 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Merzbacher, E.: Quantum Mechanics. Wiley, New York (1970)zbMATHGoogle Scholar
  14. 14.
    Mazziotti, A., Parr, R.G., Simons, G.: Regional stationary principles and regional virial theorems. J. Chem. Phys. 59, 939 (1973)CrossRefGoogle Scholar
  15. 15.
    Molahajloo, S.: The virial theorem for a class of singular pseudo-differential operators on R\(^{n}\). J. Pseudo Differ. Oper. 6, 187–196 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc. 45, 99–124 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schiff, L.I.: Quantum Mechanics, 2nd edn. McGraw-Hill Book Company Inc, New York (1955)zbMATHGoogle Scholar
  18. 18.
    Weyl, H.: The Theory of Groups and Quantum Mechanics. E. P. Dutton and Co., New York (1928)zbMATHGoogle Scholar
  19. 19.
    Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)CrossRefzbMATHGoogle Scholar
  20. 20.
    Wilcox, R.M.: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 962–982 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wong, M.W.: Weyl Transforms. Springer, New York (1998)zbMATHGoogle Scholar
  22. 22.
    Wong, M.W.: An Introduction to Pseudo-Differential Operators. World Scientific, Singapore (2014)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Physics, Hunter College and Graduate CenterCity University of New YorkNew YorkUSA

Personalised recommendations