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On Close Relationship Between Classical Time-Dependent Harmonic Oscillator and Non-relativistic Quantum Mechanics in One Dimension

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Abstract

In this paper, I present a mapping between representation of some quantum phenomena in one dimension and behavior of a classical time-dependent harmonic oscillator. For the first time, it is demonstrated that quantum tunneling can be described in terms of classical physics without invoking violations of the energy conservation law at any time instance. A formula is presented that generates a wide class of potential barrier shapes with the desirable reflection (transmission) coefficient and transmission phase shift along with the corresponding exact solutions of the time-independent Schrödinger’s equation. These results, with support from numerical simulations, strongly suggest that two uncoupled classical harmonic oscillators, which initially have a 90° relative phase shift and then are simultaneously disturbed by the same parametric perturbation of a finite duration, manifest behavior which can be mapped to that of a single quantum particle, with classical ‘range relations’ analogous to the uncertainty relations of quantum physics.

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References

  1. Dragoman, D., Dragoman, M.: Quantum-Classical Analogies. Springer, Berlin (2004)

    MATH  Google Scholar 

  2. Razavy, M.: Quantum Theory of Tunneling. World Scientific, Singapore (2003)

    Book  MATH  Google Scholar 

  3. Kaiser, J.F.: On a simple algorithm to calculate the “energy” of a signal. In: Proc. IEEE ICASSP-90, Albuquerque, NM, Apr. 1990, pp. 381–384

    Google Scholar 

  4. Kaiser, J.F.: On Teager’s energy algorithm and its generalizations to continuous signals. In: Proc. 4th IEEE DSP Workshop, Mohonk (New Paltz), NY, Sept. 1990

    Google Scholar 

  5. Maragos, P., Kaiser, J.F., Quatieri, T.F.: On amplitude and frequency demodulation using energy operators. IEEE Trans. Signal Process. 41(4), 1532–1550 (1993)

    Article  MATH  ADS  Google Scholar 

  6. Maragos, P., Kaiser, J.F., Quatieri, T.F.: Energy separation in signal modulations with applications to speech analysis. IEEE Trans. Signal Process. 41, 3024–3051 (1993)

    Article  MATH  ADS  Google Scholar 

  7. Ermakov, V.P.: Univ. Izv. Kiev 20, 1 (1880)

    Google Scholar 

  8. Pinney, E.: Proc. Am. Math. Soc. 1, 681 (1950)

    MATH  MathSciNet  Google Scholar 

  9. Ovsienko, V., Tabachnikov, S.: Projective Differential Geometry Old and New. Cambridge University Press, Cambridge (2005). ISBN 0-521-83186-5

    MATH  Google Scholar 

  10. Buckingham, E.: On physical similar systems, illustrations of the use of dimensional equations. Phys. Rev. 4, 345–376 (1914)

    Article  ADS  Google Scholar 

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    Alexander Davydov

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  1. Alexander Davydov
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Correspondence to Alexander Davydov.

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Davydov, A. On Close Relationship Between Classical Time-Dependent Harmonic Oscillator and Non-relativistic Quantum Mechanics in One Dimension. Int J Theor Phys 50, 1451–1467 (2011). https://doi.org/10.1007/s10773-010-0654-1

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