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Foundations of Physics

, Volume 35, Issue 4, pp 605–625 | Cite as

On the Classical Limit in Bohm’s Theory

  • Gary E. BowmanEmail author
Article

Abstract

The standard means of seeking the classical limit in Bohmian mechanics is through the imposition of vanishing quantum force and quantum potential for pure states. We argue that this approach fails, and that the Bohmian classical limit can be realized only by combining narrow wave packets, mixed states, and environmental decoherence.

Keywords

Bohmian mechanics classical limit 

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References

  1. Holland, P. 1993The Quantum Theory of MotionCambridge University PressCambridgeGoogle Scholar
  2. Cushing, J. 1994Quantum Mechanics: Historical Contingency and the Copenhagen HegemonyUniversity of Chicago PressChicagoGoogle Scholar
  3. Bohm, D., Hiley, B. 1993The Undivided Universe: An Ontological Interpretation of Quantum TheoryRoutledgeLondonGoogle Scholar
  4. Cushing, J., Bowman, G. 1999Bohmian mechanics and chaosButterfield, J.Pagonis, C. eds. From Physics to PhilosophyCambridge University PressCambridgeGoogle Scholar
  5. Dickson, W. 1998Quantum Chance and Non-LocalityCambridge University PressCambridgechap. 5.Google Scholar
  6. Holland, P., Kyprianidis, A. 1988Quantum potential, uncertainty and the classical limitAnn. Inst. Henrí Poincaré49325Google Scholar
  7. Home, D. 1997Conceptual Foundations of Quantum PhysicsPlenumNew Yorkchap. 3.Google Scholar
  8. Ballentine, L. 1998Quantum Mechanics: A Modern DevelopmentWorld ScientificSingaporechap. 14Google Scholar
  9. Makowski, A. 2003Exact classical limit of quantum mechanics noncentral potentials and Ermakov-type invariantsPhys. Rev. A68022102Google Scholar
  10. Makowski, A. 2002Exact classical limit of quantum mechanics central potentials and specific statesPhys. Rev. A65032103Google Scholar
  11. Makowski, A. 1999Potentials for identical classical and quantum motionsPhys.Lett.A25883Google Scholar
  12. Makowski, A., Górska, K. 2002Bohr’s correspondence principle the cases for which it is exactPhys. Rev. A66062103Google Scholar
  13. Makowski, A., Konkel, S. 1998Identical motion in classical and quantum mechanicsPhys.Rev. A58497500Google Scholar
  14. Konkel, S., Makowski, A. 1998Regular and chaotic causal trajectories for the Bohm potential in a restricted spacePhys. Lett. A2389500Google Scholar
  15. Drr, D., Goldstein, S., Zanghi, N. 1992Quantum chaos, classical randomness, and Bohmian mechanics.J. Stat. Phys.6825900Google Scholar
  16. G. Bowman, Wave packets, quantum chaos and the classical limit of Bohmian mechanics, Ph.D. thesis, University of Notre Dame (2000).Google Scholar
  17. Bowman, G. 2002Wave packets and Bohmian mechanics in the kicked rotator.Phys. Lett. A298700Google Scholar
  18. Lan, B. 2000Violation of the correspondence principle breakdown of the Bohm–Newton trajectory correspondence in a macroscopic systemPhys. Rev. A61032105Google Scholar
  19. Schwengelbeck, U., Faisal, F. 1995Definition of Lyapunov exponents and KS entropy in quantum dynamicsPhys. Lett. A199281Google Scholar
  20. Falsaperla, P., Fonte, G. 2003On the motion of a single particle near a nodal line in the de Broglie-Bohm interpretation of quantum mechanics.Phys. Lett. A31638200Google Scholar
  21. Frisk, H. 1997Properties of the trajectories in Bohmian mechanicsPhys. Lett. A22713900Google Scholar
  22. Z. Malik and C. Dewdney, ‘‘Quantum mechanics, chaos, and the Bohm theory’’, quant-ph/9506026.Google Scholar
  23. Dewdney, C., Malik, Z. 1996Measurement, decoherence and chaos in quantum pinballPhys. Lett. A22018300Google Scholar
  24. Sengupta, S., Chattaraj, P. 1996The quantum theory of motion and signatures of chaos in the quantum behaviour of a classically chaotic systemPhys. Lett. A21511900Google Scholar
  25. Faisal, F., Schwengelbeck, U. 1995Unified theory of Lyapunov exponents and a positive example of deterministic quantum chaosPhys. Lett. A2073100Google Scholar
  26. Iacomelli, G., Pettini, M. 1996Regular and chaotic quantum motionsPhys. Lett. A2122900Google Scholar
  27. Bonfim, O., Florencio, J., Barreto, F.S. 2000Chaotic Bohm’s trajectories in a quantum circular billiardPhys. Lett. A27712900Google Scholar
  28. Bonfim, O., Florencio, J., Barreto, F.S. 1998Quantum chaos in a double square well an approach based on Bohm’s view of quantum mechanicsPhys. Rev. E586851Google Scholar
  29. Bonfim, O., Florencio, J., Barreto, F.S. 1998Chaotic dynamics in billiards using {B}ohm’s quantum mechanicsPhys. Rev. E 58R2693Google Scholar
  30. Makowski, A., Peplowski, P., Dembiński, S. 2000Chaotic causal trajectories the role of the phase of stationary statesPhys. Lett. A266241Google Scholar
  31. Parmenter, R., Valentine, R. 1997Chaotic causal trajectories associated with a single stationary state of a system of noninteracting particlesPhys. Lett. A227500Google Scholar
  32. Parmenter, R., Valentine, R. 1995Deterministic chaos and the causal interpretation of quantum mechanicsPhys. Lett. A201100Google Scholar
  33. dePolavieja, G. 1996Exponential divergence of neighboring quantal trajectoriesPhys. Rev. A53205900Google Scholar
  34. Schwengelbeck, U., Faisal, F. 1997Transition to deterministic chaos in a periodically driven quantum system and breaking of the time-reversal symmetryPhys. Rev. E55626000Google Scholar
  35. de Sales, J., Florencio, J. 2001Bohmian quantum trajectories in a square billiard in the bouncing ball regimePhysica A29010100Google Scholar
  36. V. Allori, D. Dürr, S. Goldstein, and N. Zanghi, ‘‘Seven steps towards the classical world’’, (2001). {\tt quant-ph/0112005}.Google Scholar
  37. Ballentine, L. 1996The emergence of classical properties from quantum mechanicsClifton, R. eds. Perspectives on Quantum Reality: Non-Relativistic, Relativistic and Field-TheoreticKluwerDordrechtGoogle Scholar
  38. Appleby, D. 1999Generic Bohmian trajectories of an isolated particleFound. Phys.29186300Google Scholar
  39. Appleby, D. 1999Bohmian trajectories post-decoherenceFound. Phys.29188500Google Scholar
  40. Holland, P. 1996Is quantum mechanics universalCushing, J.Fine, A.Goldstein, S. eds. Bohmian Mechanics and Quantum Theory: An AppraisalKluwerDordrechtGoogle Scholar
  41. Schiff, L., Mechanics, Quantum 19683rd edn.McGraw-HillNew YorkGoogle Scholar
  42. Lichtenberg, A., Lieberman, M. 1992Regular and Chaotic DynamicsSpringer-VerlagNew YorkGoogle Scholar
  43. Izrailev, F. 1990Simple models of quantum chaos spectrum and eigenfunctions Phys. Rep196299Google Scholar
  44. G. Casati, B. Chirikov, F. Izraelev, and J. Ford, “Stochastic behavior of a quantum pendulum under a periodic perturbation,” in “Stochastic Behavior in Classical and Quantum Hamiltonian Systems,” G. Casati and J. Ford, eds., (Springer-Verlag, New York, 1979).Google Scholar
  45. Lan, B. 2001Bohm’s quantum-force time series stable distribution, flat power spectrum, and implicationPhys. Rev. A63042105Google Scholar
  46. Messiah, A. 1964Quantum MechanicsNorth-HollandAmsterdam216218Google Scholar
  47. Jackson, E. 1991Perspectives of Nonlinear Dynamics 1Cambridge University PressCambridgeGoogle Scholar
  48. Tegmark, M. 1993Apparent wave function collapse caused by scatteringFound. Phys. Lett.657100Google Scholar
  49. Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I., Zeh, H. 1996Decoherence and the Appearance of a Classical World in Quantum TheorySpringer-VerlagNew YorkGoogle Scholar
  50. Ford, J., Mantica, G. 1992Does quantum mechanics obey the correspondence principleIs it complete?’’ Am. J. Phys.60108600Google Scholar
  51. Zurek, W., Paz, J. 1994Decoherence, chaos, and the second lawPhys. Rev. Lett.72250800Google Scholar
  52. Casati, G., Chirikov, B. 1995Comment on Decoherence, chaos, and the second law’Phys. Rev. Lett.7535000Google Scholar
  53. Zurek, W., Paz, J. 1995Reply to Casati and ChirikovPhys. Rev. Lett.7535100Google Scholar
  54. Hilborn, R. 2000Chaos and Nonlinear Dynamics, 2nd edn.Oxford University PressOxfordchap. 12Google Scholar
  55. d’Espagnat, B. 1995Veiled Reality: An Analysis of Present-Day Quantum Mechanical ConceptsAddison-WesleyReading, MAGoogle Scholar
  56. Blum, K. 1996Density Matrix Theory and ApplicationsPlenumNew YorkGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Physics and AstronomyNorthern Arizona UniversityFlagstaffUSA

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