Pseudo-Differential Operators in an Operational Model of the Quantum Measurement of Observables

  • Leonid Sevastyanov
  • Alexander Zorin
  • Alexander Gorbachev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7125)


The measurement procedure transforms an isolated (closed) quantum system into an open one. The operators of observables of a rather simple explicit form are converted into pseudo-differential operators of more complicated form. A stable numerical method for studying the discrete spectra of the measured observables on the basis of the observed discrete spectra of the expected observables is developed. Thus, a method for establishing a correspondence between theoretical data in the conventional quantum mechanics of (isolated) quantum objects and the experimental data on their measured values for open quantum objects is proposed.


operational model quantum measurement Ritz functional Ritz matrix stable numerical method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berezin, F.A.: On a representation of operators with the help of functionals. Trans. Moscow Math. Soc. 17, 117–196 (1967)Google Scholar
  2. 2.
    Berezin, F.A.: Quantization. Math. USSR-Izv. 8(5), 1109–1165 (1974)CrossRefzbMATHGoogle Scholar
  3. 3.
    Berezin, F.A., Shubin, M.A.: Lectures on quantum mechanics. MSU Publ., Moscow (1972)Google Scholar
  4. 4.
    Blokhintzev, D.I.: The Gibbs Quantum Ensemble and its Connection with the Classical Ensemble. Journ. of Phys. II(1), 71–74 (1940)MathSciNetGoogle Scholar
  5. 5.
    Blokhintzev, D., Nemirovsky, P.: Connection of the Quantum Ensemble with the Gibbs Classical Ensemble. II. Journ. of Phys. III(3), 191–194 (1940)Google Scholar
  6. 6.
    Braginsky, V.B., Vorontsov, Y.I., Halily, F.Y.: Optimal quantum measurements in detectors of gravitational radiation. Lett. Journ. Exper. Theor. Phys. 27, 296–301 (1978)Google Scholar
  7. 7.
    Braginsky, V.B., Vorontsov, Y.I., Halily, F.Y.: Quantum features of the ponderomotive meter of electromagnetic energy. Journ. Exper. Theor. Phys. 73, 1340–1343 (1977)Google Scholar
  8. 8.
    Cohen, L.: Generalized Phase-Space Distribution Functions. J. Math. Phys. 7(5), 781–786 (1966)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Davies, E.B.: Quantum Theory of Open Systems. Acad. Press Inc. (1976)Google Scholar
  10. 10.
    Gelfand, I.M., Vilenkin, N.J.: Generalized Functions. Some Applications of Harmonic Analysis. Rigged Hilbert Spaces, vol. 4. Academic Press, New York (1964)Google Scholar
  11. 11.
    Gracia-Bondía, J.M., Varilly, J.C.: Nonnegative mixed states in Weyl-Wigner-Moyal theory. Phys. Lett. A 128, 20–24 (1988)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Holevo, A.S.: On the principle of quantum nondemolition measurements. Theor. Math. Phys. 65(3), 1250–1255 (1985)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland (1982)Google Scholar
  14. 14.
    Holevo, A.S.: Statistical Structure of Quantum Theory. Lect. Notes Phys., vol. m67. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kuryshkin, V.V.: Is it possible to reduce a quantum mechanics to statistical theory? In: Book: Mathematics, Mechanics, Phys., vol. (6), pp. 211–215. PFU Publ., Moscow (1969)Google Scholar
  16. 16.
    Kuryshkin, V.V.: On the construction of quantum operators. Izv. VUZov. Phys. (11), 102–106 (1971)Google Scholar
  17. 17.
    Kuryshkin, V. V.: Quantum Distribution Functions. PhD Diss. Moscow PFU (1969)Google Scholar
  18. 18.
    Mehta, C.L.: Phase-Space Formulation of the Dynamics of Canonical Variables. J. Math. Phys. 5(5), 677–686 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mihlin, S.G.: Numerical realization of variational methods. Nauka Publ., Moscow (1966)Google Scholar
  20. 20.
    Ozawa, M.: Quantum measuring processes of continuous observables. J. Math. Phys. 25, 79–87 (1984)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Analysis of Operators, vol. IV. Acad. Press, N.-Y (1978)zbMATHGoogle Scholar
  22. 22.
    Richtmyer, R.D.: Principles of Advanced Mathematical Physics, vol. 1. Springer, New York (1978)CrossRefzbMATHGoogle Scholar
  23. 23.
    Sevastyanov, L.A., Zorin, A.V.: The method of lower bounds for the eigenvalues of the Hamiltonian differential operator in quantum mechanics of Kuryshkin. Bull. PFUR, Ser. Appl. And Comp. Math. 1(1), 134–144 (2002)Google Scholar
  24. 24.
    Terletsky, Y.P.: The limiting transition from quantum to classical mechanics. Journ. Exper. Theor. Phys. 7(11), 1290–1298 (1937)Google Scholar
  25. 25.
    Von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton Univ. Press, Princeton (1955)zbMATHGoogle Scholar
  26. 26.
    Weyl, H.: Quantenmechanik und Gruppentheorie. Zeitschrift für Physik 46, 1–46 (1927)CrossRefzbMATHGoogle Scholar
  27. 27.
    Wigner, E.: On the Quantum Correction For Thermodynamic Equilibrium. Phys. Rev. 40(5), 749–759 (1932)CrossRefzbMATHGoogle Scholar
  28. 28.
    Wódkiewicz, K.: Operational Approach to Phase-Space Measurements in Quantum Mechanics. Phys. Rev. Lett. 52, 1064 (1984)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zorin, A.V.: Approximate determination of states in quantum mechanics of Kuryshkin. Bull. PFUR, Ser. Physics (12), 81–87 (2004)Google Scholar
  30. 30.
    Zorin, A.V.: The method of study essential and discrete spectra of the Hamiltonian in quantum mechanics of Kuryshkin. Bull. PFUR, Ser. Appl. And Comp. Math. 3(1), 121–131 (2004)MathSciNetGoogle Scholar
  31. 31.
    Zorin, A.V., Kuryshkin, V.V., Sevastyanov, L.A.: Description of the spectrum of a hydrogen-like atom. Bull. Of PFUR. Ser. Phys. 6(1), 62–66 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Leonid Sevastyanov
    • 1
  • Alexander Zorin
    • 1
  • Alexander Gorbachev
    • 1
  1. 1.Peoples’ Friendship University of RussiaRussia

Personalised recommendations