Skip to main content
SpringerLink
Log in

Contribution to understanding the mathematical structure of quantum mechanics

  • Quantum Optics and Fundamentals of Quantum Mechanics
  • Published:
Optics and Spectroscopy Aims and scope Submit manuscript

Abstract

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, the Born rule, commutation and uncertainty relations, probability density current, momentum operator, and rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, the Schrödinger equation, and the Dirac equation are obtained from the requirement of the relativistic invariance of the space-time Fisher information. The limit case of th e δ-like probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Manyparticle systems and the postulates of quantum mechanics are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. O. Scully, B. G. Englert, and H. Walther, Nature 351, 111 (1991).

    Article  ADS  Google Scholar 

  2. A. Zeilinger, Rev. Mod. Phys. 71, S288 (1999).

    Article  Google Scholar 

  3. A. Zeilinger, G. Weihs, T. Jennewein, and M. Aspelmeyer, Nature 433, 230 (2005).

    Article  ADS  Google Scholar 

  4. M. Arndt, O. Nairz, and A. Zeilinger, in Quantum [Un]speakables. From Bell to Quantum Information, Ed. by R. A. Bertlmann and A. Zeilinger (Springer, Berlin, 2002).

    Google Scholar 

  5. G. Birkhoff and J. von Neumann, Ann. Math. 37, 743 (1936).

    Article  Google Scholar 

  6. R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).

    Article  ADS  MathSciNet  Google Scholar 

  7. D. Bohm, Phys. Rev. 85, 166 (1952).

    Article  ADS  MathSciNet  Google Scholar 

  8. D. Bohm, Phys. Rev. 85, 180 (1952).

    Article  ADS  MathSciNet  Google Scholar 

  9. D. Bohm, Phys. Rev. 89, 458 (1952).

    Article  ADS  MathSciNet  Google Scholar 

  10. H. Everett III, Rev. Mod. Phys. 29, 454 (1957).

    Article  ADS  MathSciNet  Google Scholar 

  11. J.A. Wheeler, Rev. Mod. Phys. 29, 463 (1957).

    Article  ADS  MathSciNet  Google Scholar 

  12. G. Ludwig, Commun. Math. Phys. 9, 1 (1968).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. A. Landé, Am. J. Phys. 42, 459 (1974).

    Article  ADS  Google Scholar 

  14. J. G. Cramer, Rev. Mod. Phys. 58, 647 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  15. B. Roy Frieden, J. Mod. Opt. 35, 1297 (1988); B. Roy Frieden, Am. J. Phys. 57, 1004 (1989); B. Roy Frieden and B. H. Soffer, Phys. Rev. E 52, 2274 (1995); B. Roy Frieden, Physics from Fisher Information (Cambridge University Press, Cambridge, 1998).

    Article  Google Scholar 

  16. D. I. Fivel, Phys. Rev. A 50, 2108 (1994).

    Article  ADS  Google Scholar 

  17. A. Bohr and O. Ulfbeck, Rev. Mod. Phys. 67, 1 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  18. Č. Brukner and A. Zeilinger, Phys. Rev. Lett. 83, 3354 (1999).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. S. M. Roy and V. Singh, Phys. Lett. A 255, 201 (1999).

    Article  ADS  Google Scholar 

  20. Ch. A. Fuchs and A. Peres, Physics Today 70 (2000).

  21. W. E. Lamb, Am. J. Phys. 69, 413 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  22. L. Hardy, arXiv:quant-ph/0111068 v1 (2001).

  23. J. Summhammer, arXiv:quant-ph/0102099 v1 (2001).

  24. R. R. Parwani, arXiv:hep-th/0401190 v2 (2004).

  25. F. Laloé, Am. J. Phys. 69, 655 (2001).

    Article  ADS  Google Scholar 

  26. L. Skala and V. Kapsa, Physica E 29, 119 (2005); L. Skala and V. Kapsa, Collect. Czech. Chem. Commun. 70, 621 (2005).

    Article  ADS  Google Scholar 

  27. A. E. Allahverdyan, R. Balian, and Th. M. Nieuwenhuizen, Europhys. Lett. 61, 452 (2003).

    Article  ADS  Google Scholar 

  28. A. E. Allahverdyan, R. Balian, and Th. M. Nieuwenhuizen, arXiv:cond-mat/0408316 (2004).

  29. O. Hay and A. Peres, Phys. Rev. A 58, 116 (1998).

    Article  ADS  Google Scholar 

  30. M. Born, Z. Phys. 38, 803 (1926).

    Article  ADS  Google Scholar 

  31. M. Ozawa, Phys. Rev. Lett. 88, 050402-1 (2002).

    Google Scholar 

  32. E. Madelung, Z. Phys. 40, 322 (1926).

    ADS  Google Scholar 

  33. A. S. Davydov, Quantum Mechanics (Pergamon Press, New York, 1976).

    Google Scholar 

  34. R. Shankar, Principles of Quantum Mechanics (Plenum Press, New York and London, 1994).

    MATH  Google Scholar 

  35. W. Heisenberg, Z. Phys. 43, 172 (1927).

    Article  ADS  Google Scholar 

  36. S. L. Braunstein, C. M. Caves, and G. J. Milburn, Annals of Physics 247, 135 (1996).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. D. C. Brody and B. K. Meister, J. Phys. A 32, 4921 (1999).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. M. J. W. Hall, Phys. Rev. A 64, 052103 (2001).

    Google Scholar 

  39. M. Reginatto, Phys. Rev. A 58, 1775 (1998); Erratum ibid. A 60, 1730 (1999).

    Article  ADS  Google Scholar 

  40. R. A. Fisher, Proc. Cambr. Phil. Soc. 22, 700 (1925).

    MATH  Google Scholar 

  41. T. Cover and J. Thomas, Elements of Information Theory (Wiley, New York, 1991).

    MATH  Google Scholar 

  42. M. J. W. Hall, Phys. Rev. A 62, 012107 (2000).

    Google Scholar 

  43. D. N. Klyshko, Phys. Lett. A 243, 179 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. P. Busch, Found. Phys. 20, 1 (1990); P. Busch, Found. Phys. 20, 33 (1990).

    Article  ADS  Google Scholar 

  45. J. Hilgerwood, Am. J. Phys. 70, 301 (2002).

    Article  ADS  Google Scholar 

  46. Einstein-Sommerfeld Briefwechsel (Schwabe, Basel, 1968).

  47. F. H. Fröhner, Z. Naturforsch. 53a, 637 (1998).

    Google Scholar 

  48. A. Zeilinger, Found. Phys. 29, 631 (1999).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, 121 16, Prague 2, Czech Republic

    L. Skála & V. Kapsa

  2. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    L. Skála

Authors
  1. L. Skála
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Kapsa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Skála.

Additional information

The text was submitted by the authors in English.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Skála, L., Kapsa, V. Contribution to understanding the mathematical structure of quantum mechanics. Opt. Spectrosc. 103, 434–450 (2007). https://doi.org/10.1134/S0030400X07090135

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0030400X07090135

PACS numbers

Search

Navigation