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Exact classical limit of the quantum bouncer

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Abstract

In this paper, we develop a systematic approach to determine the classical limit of periodic quantum systems and we applied it successfully to the problem of the quantum bouncer. It is well known that, for periodic systems, the classical probability density does not follow the quantum probability density. Instead, it follows the local average in the limit of large quantum numbers. Guided by this fact, and expressing both the classical and quantum probability densities as Fourier expansions, here we show that local averaging implies that the Fourier coefficients approach each other in the limit of large quantum numbers. The leading term in the quantum Fourier coefficient yields the exact classical limit, but subdominant terms also emerge, which we may interpret as quantum corrections at the macroscopic level. We apply this theory to the problem of a particle bouncing under the gravity field and show that the classical probability density is exactly recovered from the quantum distribution. We show that for realistic systems, the quantum corrections are strongly suppressed (by a factor of \(\sim 10^{-10}\)) with respect to the classical result.

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References

  1. A.J. Makowski, A brief survey of various formulations of the correspondence principle. Eur. J. Phys. 27, 1133–1139 (2006)

    Article  MathSciNet  Google Scholar 

  2. R.L. Liboff, The correspondence principle revisited. Phys. Today 37, 50–55 (1984)

    Article  Google Scholar 

  3. G.Q. Hassoun, D.H. Kobe, Synthesis of the Planck and Bohr formulations of the correspondence principle. Am. J. Phys. 57, 658–662 (1989)

    Article  ADS  Google Scholar 

  4. G. Wentzel, Eine verallgemeinerung der quantenbedingungen für die zwecke der wellenmechanik. Z. Phys. 38, 518–529 (1926)

    Article  MATH  ADS  Google Scholar 

  5. H.A. Kramers, Wellenmechanik und halbzahlige quantisierung. Z. Phys. 39, 828–840 (1926)

    Article  MATH  ADS  Google Scholar 

  6. L. Brillouin, La mécanique ondulatoire de Schrödinger; une méthode générale de resolution par approximations successives. Compt. Rend. Hebd. Seances Acad. Sci. 183, 24–26 (1926)

    MATH  Google Scholar 

  7. R.P. Feynman, A.R. Hibbs, Quantum mechanics and path integrals, International series in pure and applied physics (McGraw-Hill, New York, NY, 1965)

    MATH  Google Scholar 

  8. P. Ehrenfest, Bemerkung über die angenäherte gültigkeit der klassischen mechanik innerhalb der quantenmechanik. Z. Phys. 45, 455–457 (1927)

    Article  MATH  ADS  Google Scholar 

  9. L.E. Ballentine, Y. Yang, J.P. Zibin, Inadequacy of Ehrenfest’s theorem to characterize the classical regime. Phys. Rev. A 50, 2854–2859 (1994)

    Article  ADS  Google Scholar 

  10. R.L. Liboff, Introductory Quantum Mechanics Addison Wesley (Addison-Wesley, Reading, MA, 1980)

    Google Scholar 

  11. R. Robinett, R.W. Robinett, Quantum mechanics: Classical results, modern systems, and visualized examples (Oxford University Press, 2006)

  12. R.W. Robinett, Quantum and classical probability distributions for position and momentum. Am. J. Phys. 63, 823–832 (1995)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  13. M.A. Doncheski, R.W. Robinett, Comparing classical and quantum probability distributions for an asymmetric infinite well. Eur. J. Phys. 21, 217–228 (2000)

    Article  Google Scholar 

  14. R.W. Robinett, Visualizing the solutions for the circular infinite well in quantum and classical mechanics. Am. J. Phys. 64, 440–446 (1996)

    Article  ADS  Google Scholar 

  15. R.W. Robinett, Visualizing classical and quantum probability densities for momentum using variations on familiar one-dimensional potentials. Eur. J. Phys. 23, 165–174 (2002)

    Article  Google Scholar 

  16. G. Yoder, Using classical probability functions to illuminate the relation between classical and quantum physics. Am. J. Phys. 74, 404–411 (2006)

    Article  ADS  Google Scholar 

  17. C. Semay, L. Ducobu, Quantum and classical probability distributions for arbitrary hamiltonians. Eur. J. Phys. 37, 045403 (2016)

    Article  MATH  Google Scholar 

  18. C. Leubner, M. Alber, N. Schupfer, Critique and correction of the textbook comparison between classical and quantum harmonic oscillator probability densities. Am. J. Phys. 56, 1123–1129 (1988)

    Article  ADS  Google Scholar 

  19. J. Bernal, A. Martín-Ruiz, J. García-Melgarejo, A simple mathematical formulation of the correspondence principle. J. Mod. Phys. 4, 108 (2013)

    Article  Google Scholar 

  20. A. Martín-Ruiz, J. Bernal, A. Carbajal-Dominguez, Macroscopic quantum behaviour of periodic quantum systems. J. Mod. Phys. 5, 44 (2013)

    Article  Google Scholar 

  21. A. Martín-Ruiz, J. Bernal, A. Frank, A. Carbajal-Dominguez, The classical limit of the quantum Kepler problem. J. Mod. Phys. 4, 818 (2013)

    Article  Google Scholar 

  22. R. Durrett, Probability: Theory and Examples, 2nd edn. (Duxbury Press, Belmont, CA, 1996)

    MATH  Google Scholar 

  23. A. Leon-Garcia, Probability, Statistics, and Random Processes for Electrical Engineering, 3rd edn. (Pearson/Prentice Hall, Upper Saddle River, NJ, 2008)

    Google Scholar 

  24. W.B. Case, Wigner functions and weyl transforms for pedestrians. Am. J. Phys. 76, 937–946 (2008)

    Article  ADS  Google Scholar 

  25. M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Distribution functions in physics: fundamentals. Phys. Rep. 106, 121–167 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  26. J. Mostowski, J. Pietraszewicz, Wigner function for harmonic oscillator and the classical limit, (2021)

  27. E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, Charmonium: the model. Phys. Rev. D 17, 3090–3117 (1978)

    Article  ADS  Google Scholar 

  28. C. Weisbuch, B. Vinter, Quantum Semiconductor Structures (Academic Press, San Diego, 1991)

    Book  Google Scholar 

  29. V.V. Nesvizhevsky, H.G. Börner, A.K. Petukhov, H. Abele, S. Baeßler, F.J. Rueß, T. Stöferle, A. Westphal, A.M. Gagarski, G.A. Petrov, A.V. Strelkov, Quantum states of neutrons in the Earth’s gravitational field. Nature 415, 297–299 (2002)

    Article  ADS  Google Scholar 

  30. R.L. Gibbs, The quantum bouncer. Am. J. Phys. 43, 25–28 (1975)

    Article  ADS  Google Scholar 

  31. J. Gea-Banacloche, A quantum bouncing ball. Am. J. Phys. 67, 776–782 (1999)

    Article  ADS  Google Scholar 

  32. O. Vallée, M. Soares, Airy Functions and Applications to Physics (Imperial College Press, 2004)

  33. J.R. Albright, Integrals of products of Airy functions. J. Phys. A: Math. Gen. 10, 485–490 (1977)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  34. I. S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic Press, New York, 1994) edited by A. Jeffrey and D. Zwillinger

  35. S. Singh, S.P. Suman, V.A. Singh, Quantum–classical correspondence for a particle in a homogeneous field. Eur. J. Phys. 37, 065405 (2016)

    Article  MATH  Google Scholar 

  36. G.G. Cabrera, M. Kiwi, Large quantum-number states and the correspondence principle. Phys. Rev. A 36, 2995–2998 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  37. B. Gao, Breakdown of Bohr’s correspondence principle. Phys. Rev. Lett. 83, 4225–4228 (1999)

  38. C. Eltschka, H. Friedrich, M.J. Moritz, Comment on “breakdown of Bohr’s correspondence principle’’. Phys. Rev. Lett. 86, 2693–2693 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  39. C. Boisseau, E. Audouard, J. Vigué, Comment on “breakdown of Bohr’s correspondence principle’’. Phys. Rev. Lett. 86, 2694–2694 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  40. R.L. Liboff, Bohr correspondence principle for large quantum numbers. Found. Phys. 5, 271–293 (1975)

    Article  ADS  Google Scholar 

  41. R.L. Liboff, On the potential \(x^{2N}\) and the correspondence principle. Int. J. Theor. Phys. 18, 185–191 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  42. D. Sen, S. Sengupta, Classical limit for quantum mechanical energy eigenfunctions. Curr. Sci. 87, 620–627 (2004)

    Google Scholar 

  43. D. Sen, S. Sengupta, Classical limit problem of quantum mechanical energy eigenfunctions - extension to two- and three-dimensional cases. Int. J. Mod. Phys. A 20, 7515–7524 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  44. D. Sen, S. Sengupta, A critique of the classical limit problem of quantum mechanics. Found. Phys. Lett. 19, 403–421 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Juan A. Cañas, J. Bernal, A. Martín-Ruiz, Is the equivalence principle compatible with quantum mechanics? (2022), to be submitted

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Acknowledgements

J.A.C. was supported by the CONACyT master fellowship No. 725033. A.M.-R. has been partially supported by DGAPA-UNAM Project No. IA102722 and by Project CONACyT (México) No. 428214.

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Authors and Affiliations

  1. División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, 86690, Cunduacán, Tabasco, México

    Juan A. Cañas & J. Bernal

  2. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510, Ciudad de México, México

    A. Martín-Ruiz

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  1. Juan A. Cañas
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  2. J. Bernal
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  3. A. Martín-Ruiz
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Correspondence to A. Martín-Ruiz.

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Cañas, J.A., Bernal, J. & Martín-Ruiz, A. Exact classical limit of the quantum bouncer. Eur. Phys. J. Plus 137, 1310 (2022). https://doi.org/10.1140/epjp/s13360-022-03529-2

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