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Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue

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Abstract

The correspondence principle states that the quantum system will approach the classical system in high quantum numbers. Indeed, the average of the quantum probability density distribution reflects a classical-like distribution. However, the probability of finding a particle at the node of the wave function is zero. This condition is recognized as the nodal issue. In this paper, we propose a solution for this issue by means of complex quantum random trajectories, which are obtained by solving the stochastic differential equation derived from the optimal guidance law. It turns out that point set A, which is formed by the intersections of complex random trajectories with the real axis, can represent the quantum mechanical compatible distribution of the quantum harmonic oscillator system. Meanwhile, the projections of complex quantum random trajectories on the real axis form point set B that gives a spatial distribution without the appearance of nodes, and approaches the classical compatible distribution in high quantum numbers. Furthermore, the statistical distribution of point set B is verified by the solution of the Fokker–Planck equation.

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References

  1. Bohr, N.: On the constitution of atoms and molecules, Part I. Philos. Mag. 26, 1 (1913)

    ADS  MATH  Google Scholar 

  2. Bohr, N.: On the quantum theory of line spectra. In: Van der Waerden, B.L. (ed.) Sources of quantum mechanics, pp. 95–136. North-Holland, Amsterdam (1918)

    Google Scholar 

  3. Born, M.: Quantum mechanics. In: Van der Waerden, B.L. (ed.) Sources of Quantum Mechanics, pp. 181–198. North-Holland, Amsterdam (1918)

    Google Scholar 

  4. Kramers, H.A.: Intensities of spectral lines. In: Kramers, H.A. (ed.) Collected Scientific Papers, pp. 1–108. North-Holland, Amsterdam (1956)

    Google Scholar 

  5. Heisenberg, W.: The physical principles of the quantum theory. Translated by Eckart, C. and Hoyt, F. C. p. 116. Dover, New York (1949).

  6. Bokulich, A. Three puzzles about Bohr’s correspondence principle. Philos. Sci. https://philsci-archive.pitt.edu/ (2009).

  7. Van Vleck, J.H.: The correspondence principle in the statistical interpretation of quantum mechanics. Proc. Natl. Acad. Sci. USA 14, 178–188 (1928)

    ADS  MATH  Google Scholar 

  8. Smith, E.: Quantum-classical correspondence principles for locally non-equilibrium driven system. Phys. Rev. E 77, 021109 (2008)

    ADS  MathSciNet  Google Scholar 

  9. Makri, N.: Time-dependent quantum methods for large systems. Annu. Rev. Phys. Chem. 50, 167–191 (1999)

    ADS  Google Scholar 

  10. Hnlio, A.A.: Simple explanation of the classical limit. Found. Phys. 49, 1365–1371 (2019)

    ADS  MathSciNet  Google Scholar 

  11. Schiff, L.: Quantum Mechanics, pp. 66–75. McGraw-Hill, New York (1995)

    Google Scholar 

  12. Serge, E.: From X-rays to Quarks. Freeman, San Francisco (1980)

    Google Scholar 

  13. Liboff, R.: The correspondence principle revisited. Phys.To. 37, 50 (1984)

    Google Scholar 

  14. Bohm, D.: A suggested interpretation of quantum theory in terms of ‘hidden’ variables I and II. Phys. Rev. 85, 166–193 (1952)

    ADS  MathSciNet  MATH  Google Scholar 

  15. Holland, P.R.: New trajectory interpretation of quantum mechanics. Found. Phys. 28, 881–991 (1998)

    MathSciNet  Google Scholar 

  16. Mayor, F.S., Askar, A., Rabitz, H.A.: Quantum fluid dynamics in the Lagrangian representation and applications to photo-dissociation problems. J. Chem. Phys. 111, 2423–2435 (1999)

    ADS  Google Scholar 

  17. Guantes, R., Sanz, A.S., et al.: Atom-surface diffraction: a trajectory description. Surf. Sci. Rep. 53, 199–330 (2004)

    ADS  Google Scholar 

  18. Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  19. Poirier, B.: Reconciling semiclassical and Bohmian mechanics I. Stationary states. J. Chem. Phys. 121, 4501–4515 (2004)

    ADS  Google Scholar 

  20. Matzkin, A.: Bohmian mechanics, the quantum-classical correspondence and the classical limit: the case of the square billiard. Found. Phys. 39, 903–920 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Holland, P.: Is quantum mechanics universal. In: Cushing, J., Fine, A., Goldstein, S. (eds.) Bohmian Mechanics and Quantum Theory: An Appraisal. Kluwer, Dordrecht (1996)

    Google Scholar 

  22. Bowman, G.E.: On the classical limit in Bohm’s theory. Found. Phys. 35, 605–625 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  23. Shudo, A., Ikeda, K.S.: Complex classical trajectories and chaotic tunneling. Phys. Rev. Lett. 74, 682–685 (1995)

    ADS  Google Scholar 

  24. Yang, C.D.: Quantum Hamilton mechanics: Hamilton equation of quantum motion, origin of quantum operators, and proof of quantization axiom. Ann. Phys. 321, 2876–2926 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  25. Kiran, M., John, M.V.: Tunneling in energy eigenstates and complex quantum trajectories. Quant. Stud. Math. Found. 2, 403–422 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Nanayakkara, A.: Classical trajectories of 1D complex non-Hermitian Hamiltonian systems. J. Phys. A 37, 4321–4334 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  27. Yang, C.D., Wei, C.H.: Strong chaos in one-dimensional quantum system. Chaos Sol. Frac. 37, 988–1001 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  28. Cheng, J.: Chaotic dynamics in a periodically driven spin-1 condensate. Phys. Rev. A 81, 023619–23621 (2010)

    ADS  Google Scholar 

  29. Sanz, A.S., Borondo, F., Miret-Ates, S.: Particle diffraction studied using quantum trajectories. J. Phys. Condens. Matter 14, 6109–6145 (2002)

    ADS  Google Scholar 

  30. Mostafazadeh, A.: Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102, 220402 (2009)

    ADS  Google Scholar 

  31. Yang, C.D.: Wave-particle duality in complex space. Ann. Phys. 319, 444–470 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  32. Goldfarba, Y., Tannor, D.J.: Interference in Bohmian mechanics with complex action. J. Chem. Phys. 127, 161101 (2007)

    ADS  Google Scholar 

  33. Mathew, K., John, M.V.: Interfering quantum trajectories without which-way information. Found. Phys. 47, 873–886 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  34. Chou, C.C.: Trajectory description of the quantum-classical transition for wave packet interference. Ann. Phys. 371, 437–459 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  35. Gondran, M., Gondran, A.: Numerical simulation of the double slit interference with ultracold atoms. Am. J. Phys. 73, 507–532 (2005)

    ADS  Google Scholar 

  36. Sanz, A.S., et al.: Understanding interference experiments with polarized light through photon trajectories. Ann. Phys. 325, 763–784 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  37. Juffmann, T., et al.: Real-time single-molecule imaging of quantum interference. Nat. Nano. Technol. 7, 297–300 (2012)

    ADS  Google Scholar 

  38. Yang, C.D., Su, K.C.: Reconstructing interference fringes in slit experiments by complex quantum trajectories. Int. J. Quant. Chem. 113, 1253–1263 (2013)

    ADS  Google Scholar 

  39. Poirier, B.: Flux continuity and probability conservation in complexified Bohmian mechanics. Phys. Rev. A 77, 022114 (2008)

    ADS  MathSciNet  Google Scholar 

  40. John, M.V.: Probability and complex quantum trajectories: finding the missing links. Ann. Phys. 325, 2132–2139 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  41. John, M.V.: Probability and complex quantum trajectories. Ann. Phys. 324, 220–231 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  42. John, M.V.: Coherent states and modified de Broglie–Bohm complex quantum trajectories. Found. Phys. 43, 859–871 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  43. Bender, C.M., Hook, D.W., Meisinger, P.N., Wang, Q.H.: Complex correspondence principle. Phys. Rev. Lett. 104, 061601 (2010)

    ADS  Google Scholar 

  44. Barkay, H., Moiseyev, N.: Complex density probability in non-Hermitian quantum mechanics: interpretation and a formula for resonant tunneling probability amplitude. Phys. Rev. A 64, 044702 (2001)

    ADS  Google Scholar 

  45. Bender, C.M., Hook, D.W., Meisinger, P.N., Wang, Q.H.: Probability density in the complex plane. Ann. Phys. 325, 2332–2362 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  46. Bohm, D., Hiley, J.: Non-locality and locality in the stochastic interpretation of quantum mechanics. Phys. Rep. 172, 93–122 (1989)

    ADS  MathSciNet  Google Scholar 

  47. Yang, C.D., Cheng, L.L.: Optimal guidance law in quantum mechanics. Ann. Phys. 338, 167–185 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  48. Yang, C.D.: Modeling quantum harmonic oscillator in complex domain. Chao Soli Frac. 30, 342–362 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Tsung-Lien Ko and Yang-Hsuan Lin for performing the numerical simulations.

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Authors and Affiliations

  1. Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan, ROC

    Ciann-Dong Yang

  2. Department of Applied Physics, National University of Kaohsiung, Kaohsiung, Taiwan, ROC

    Shiang-Yi Han

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  1. Ciann-Dong Yang
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  2. Shiang-Yi Han
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Correspondence to Shiang-Yi Han.

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Yang, CD., Han, SY. Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue. Found Phys 50, 960–976 (2020). https://doi.org/10.1007/s10701-020-00363-3

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