Are quantum trajectories suitable for semiclassical approximations?
Source abstract
Abstract
The quantum trajectories in the de Broglie-Bohm formulation of quantum mechanics depend on an additional quantum potential derived from the full wave solution of Schrödinger's equation. The task of supplying collectively all the correct quantum results strongly alters the characteristics of the corresponding classical trajectories, which underlie semiclassical approximations to the evolving wave function. Both classical and quantum trajectories are here considered to be conservative with no influence of an external environment, even though this is the source of eventual classicality in quantum systems, that is, decoherence. The concept of integrability, closely correspondent in classical and quantum mechanics, is not preserved by the quantum trajectories. General systems, in which classical chaotic motion participates, are much harder to treat semiclassically, but quantum trajectories can be chaotic even for integrable systems. This discrepancy between the character of classical and quantum trajectories in the de Broglie-Bohm interpretation does not clarify the singular classical-quantum transition.
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Research context
OpenAlex topics: Spectral Theory in Mathematical Physics · Quantum Mechanics and Applications · Quantum chaos and dynamical systems
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
Centro Brasileiro de Pesquisas Fisicas
Funding
CNPq
References (39)
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1. D. Bohm, A suggested interpretation of the quantum theory in terms of “Hidden” variables. I, Phys. Rev. 85 (1952), p. 166.10.1103/PhysRev.85.166 ↗
2. D. Bohm, A suggested interpretation of the quantum theory in terms of “Hidden” variables. II, Phys. Rev. 85 (1952), p. 180.10.1103/PhysRev.85.180 ↗
3. L. de Broglie in Solvay Conference – 1927 Gauthier Villars Paris: La nouvelle dynamique de quanta 1928.
4. J.S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Physique Fizika 1 (1964), p. 195. (see also [5]).10.1103/PhysicsPhysiqueFizika.1.195 ↗
5. J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge, 1993.
6. M.V. Berry Chaos and Quantum Physics in Les Houches LII M.-J. Giannoni A. Voros J. Zinn-Justin ed. Amsterdam.
7. D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H.D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin, 1996.10.1007/978-3-662-03263-3 ↗
8. A. R. F. Neto, A. M. O. de Almeida, and O. Brodier, Exact Markovian evolution of quantum systems with several degrees of freedom: Phase space representations, Phys. Rev. E 108 (2023), p. 044106.10.1103/PhysRevE.108.044106 ↗
9. N. Bohr, LXXIII. On the constitution of atoms and molecules, Philos. Mag. 26 (1913), pp. 857–875.10.1080/14786441308635031 ↗
10. A. Sommrfeld, Zur quantentheorie der spektrallinien, Ann. Phys. (Leipzig) 356 (1916), pp. 1–94.10.1002/andp.19163561702 ↗
11. R.P. Feynman, Space-Time approach to non-relativistic quantum mechanics, Rev. Mod. Phys 20 (1948), pp. 367–387.10.1103/RevModPhys.20.367 ↗
12. L.S. Schulman, Techniques and Applications of Path Integration, Wiley, New York, 1981.
13. C. Morette, On the definition and approximation of Feynman's path integrals, Phys. Rev. 81 (1952), pp. 848–852.10.1103/PhysRev.81.848 ↗
14. V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978.10.1007/978-1-4757-1693-1 ↗
15. O. Bohigas, M.J. Giannoni, A.M. Ozorio de Almeida, and C. Schmit, Chaotic dynamics and the GOE-GUE transition, Nonlinearity 8 (21995), pp. 203–221.10.1088/0951-7715/8/2/005 ↗
16. M.V. Berry and K.E. Mount, Semiclassical approximations in wave mechanics, Rept. Prog. Phys. 35 (11972), pp. 315–397.10.1088/0034-4885/35/1/306 ↗
17. H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris, 1892.
18. J. Ford and M. Mantica, Does quantum mechanics obey the correspondence principle? Is it complete? Am. J. Phys. 60 (121992), pp. 1086–1098.10.1119/1.16954 ↗
19. J.H. Van Vleck, The correspondence principle in the statistical interpretation of quantum mechanics, Proc. Nat. Acad. Sci. USA 14 (1928), pp. 178–188.10.1073/pnas.14.2.178 ↗
20. M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, New York, 1990.10.1007/978-1-4612-0983-6 ↗
21. G. Lando and A.M. Ozorio de Almeida, Quantum-chaotic evolution reproduced from effective integrable trajectories, Phys. Rev. Lett. 124 (2022), p. 010402.10.1103/PhysRevLett.124.010402 ↗
22. A. Matzkin, Bohmian mechanics, the quantum-classical correspondence and the classical limit: the case of the square billiard, Found. Phys. 39 (2009), pp. 903–920.10.1007/s10701-009-9304-y ↗
23. R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillations, Ann. Phys. 69 (1972), pp. 76–160.10.1016/0003-4916(72)90006-1 ↗
24. F. Haake, S. Gnutzmann, and M. Kus, Quantum Signatures of Chaos, Springer, New York, 2018.10.1007/978-3-319-97580-1 ↗
25. E. Bogomolny, Semiclassical quantization of multidimensional systems 1992 Nonlinearity 5 805. Nonlinearity 5 (1990), pp. 1055.
26. M.V. Berry and J.P. Keating, A new asymptotic representation for ζ(1/2+it) and quantum spectral determinants, Proc. R. Soc. Lond. A 437 (1992), pp. 151–173.10.1098/rspa.1992.0053 ↗
27. E. Bogomolny, Smoothed wave functions of chaotic quantum systems, Phys. D 31 (1988), pp. 169–189.10.1016/0167-2789(88)90075-9 ↗
28. E.P. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev 40 (1932), pp. 749–759.10.1103/PhysRev.40.749 ↗
29. M.V. Berry, Quantum scars of classical closed orbits in phase space, Proc. R. Soc. Lond. A 423 (1989), pp. 219–231.10.1098/rspa.1989.0052 ↗
30. M.V. Berry, Fringes decorating anticaustics in ergodic wavefunctions, Proc. R. Soc. Lond. A 424 (1989), pp. 279–288.10.1098/rspa.1989.0082 ↗
31. A.M. Ozorio de Almeida, The Weyl representation in classical and quantum mechanics, Phys. Rep.295 (1998), pp. 265–342.10.1016/S0370-1573(97)00070-7 ↗
32. M. Sieber, Wavefunctions, Green functions and expectation values in terms of spectral determinants, Nonlinearity 20 (2007), pp. 2721–2737.10.1088/0951-7715/20/11/013 ↗
33. O. Agam and S. Fishman, Quantum eigenfunctions in terms of periodic orbits of chaotic systems, J. Phys. A 26 (1993), pp. 2113–2137.10.1088/0305-4470/26/9/010 ↗
34. A.M. Ozorio de Almeida, Phase space geometry of general quantum energy transitions, J. Phys. A 57 (2024), p. 235303.10.1088/1751-8121/ad4b7a ↗
35. V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, CUP, Cambridge, 1984.
36. A. Voros, Semi-classical approximations. Ann. Inst. H. Poincaré 24A (1976), pp. 31.
37. A.M. Ozorio de Almeida and G.L. Ingold, Metaplectic sheets and caustic traversals in the Weyl representation, J. Phys. A 47 (2014), p. 105303.10.1088/1751-8113/47/10/105303 ↗
38. B. Poirier, Reconciling semiclassical and Bohmian mechanics. I. Stationary states, J. Chem. Phys. 121 (2004), pp. 4501–4515.10.1063/1.1775766 ↗
39. B. Poirier, Reconciling semiclassical and Bohmian mechanics. VI. Multidimensional dynamics, J. Chem. Phys. 129 (2008), p. 084103.10.1063/1.2969102 ↗