Pramana

, Volume 51, Issue 5, pp 585–595 | Cite as

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

Causal quantum theory

Abstract

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’

Keywords

de Broglie-Bohm quantum mechanics quantum chaos Lyapunov numbers time-reversal invariance 

PACS Nos

05.55 03.65 05.30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I C Percival,J. Phys. B6, L229 (1973)ADSGoogle Scholar
  2. [2]
    M V Berry, and M Tavor,Proc. R. Soc. London A356, 375 (1976)ADSGoogle Scholar
  3. [3]
    G Casati, B V Chirikov, F M Izrailev, and J Ford: inStochastical behaviour in classical and quantum hamiltonian systems, Lecture Notes in Physics, (Springer, Berlin, 1979) Vol. 93Google Scholar
  4. [4]
    A M Ozorio de Almeida,Hamiltonian systems: chaos and quantisation, (Cambridge University Press, Cambridge, 1988)Google Scholar
  5. [5]
    F Haake,Quantum signatures of chaos, (Springer-Verlag, Berlin, 1991)MATHGoogle Scholar
  6. [6]
    Quantum chaos: between order and disorder, edited by G Casati and B V Chirikov, (Cambridge University Press, Cambridge, 1995)Google Scholar
  7. [7]
    E J Heller,Phys. Rev. Lett. 53, 1515 (1984)CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    M Toda and K Ikeda,Phys. Lett. A124, 165 (1987)ADSMathSciNetGoogle Scholar
  9. [9]
    W Slomczyński and K Zyczkowski,J. Math. Phys. 35, 5674 (1994)CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    R Vileda Mendes, Preprint of the conference talkAdvanced topics in applied mathematics and theoretical physics -complex systems: Classical and quantum aspects CIRM (1994)Google Scholar
  11. [11]
    B Mirbach and H J Korsch,Phys. Rev. Lett. 75, 362 (1995)CrossRefADSGoogle Scholar
  12. [12]
    K Nakamura,Quantum chaos: a new paradigm of nonlinear dynamics, (Cambridge University Press, Cambridge, 1993)MATHGoogle Scholar
  13. [13]
    D Bohm,Phys. Rev. 85, 166 (1952)CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    L de Broglie,Non-linear wave mechanics: A causal interpretation, (Elsevier, Amsterdam, 1960)MATHGoogle Scholar
  15. [15]
    J S Bell,Speakable and unspeakable in quantum mechanics, (Cambridge University Press, Cambridge, 1987)Google Scholar
  16. [16]
    P R Holland,The quantum theory of motion, (Cambridge University Press, Cambridge, 1993)Google Scholar
  17. [17]
    F H M Faisal and U Schwengelbeck,Forschungszentrum Bielefeld-Bochum-Stochastik, BiBoS Nr. 680 / 12 / 94, Universität Bielefeld (1994)Google Scholar
  18. [18]
    U Schwengelbeck and F H M Faisal,Phys. Lett. A199, 281 (1995)ADSMathSciNetGoogle Scholar
  19. [19]
    U Schwengelbeck and F H M Faisal,Phys. Rev. A50, 632 (1994)ADSGoogle Scholar
  20. [20]
    U Schwengelbeck and F H M Faisal,Forschungszentrum Bielefeld-Bochum-Stochastik, BiBoS Nr. 723 / 4 / 96, Universität Bielefeld (1996)Google Scholar
  21. [21]
    S Weigert,Z. Phys. B80, 3 (1990)CrossRefMathSciNetGoogle Scholar
  22. [22]
    S Weigert,Phys. Rev. A48, 1780 (1993)ADSMathSciNetGoogle Scholar
  23. [23]
    V I Arnold and A Avez,Ergodic problems of classical mechanics, (Benjamin, Reading, MA, 1968)Google Scholar
  24. [24]
    F H M Faisal and U Schwengelbeck,Phys. Lett. A207, 31 (1995)ADSMathSciNetGoogle Scholar
  25. [25]
    J H Hannay and M V Berry,Physica D1, 267 (1980)ADSMathSciNetGoogle Scholar
  26. [26]
    J Ford, G Mantica, and G H Ristow,Physica D50, 493 (1991)ADSMathSciNetGoogle Scholar
  27. [27]
    G Strang,Linear algebra and its applications, (Academic Press, New York, 1976)MATHGoogle Scholar
  28. [28]
    V M Alekseev and M V Yakobson,Phys. Rep. 75, 287 (1981)CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 1998

Authors and Affiliations

  1. 1.Fakultät für PhysikUniversität BielefeldBielefeldGermany

Personalised recommendations