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Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators, part II: the use of mathematical fluctuation theory

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Abstract

The first part of these two companion papers has been devoted to the extension of Hausdorff moment problem to the sequences over integrals of Kronecker powers of an appropriate vector under a generating function in the kernel. The relations between this generating function and weight function properties have been investigated over there in a quite detailed manner. This second companion paper focuses on the utilization of the “mathematical fluctuation theory” amenities in the construction of approximations to the solutions of the expectation value dynamics of the quantum dynamical systems. The fluctuation freee approximation matching with the classical mechanical behaviour is followed by the first and then the second order fluctuation approximations. Beside the well known “Energy Conservation Law”s counterparts in these approximations of quantum expectation value dynamics are also presented.

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References

  1. M. Ayvaz, M. Demiralp, Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators, part I:integral representation of Kronecker power series and multivariate Hausdorff moment problems, J. Math. Chem. (2014). doi:10.1007/s10910-014-0371-8

  2. P. Ehrenfest, Bemerkungüber die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Z. Phys. 45(7–8), 455–457 (1927)

  3. M. Demiralp, No fluctuation approximation in any desired precision for univariate function matrix representations. J. Math. Chem. 47, 99–110 (2010). doi:10.1007/s10910-009-9533-5

    Article  CAS  Google Scholar 

  4. B. Tunga, M. Demiralp, Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation. J. Math. Chem. 49, 894–909 (2012). doi:10.1007/s10910-010-9786-z

    Article  Google Scholar 

  5. M. Demiralp, Various Parallel and Diversive Aspects of the Mathematical Fluctuations Theory with the Related Standing Issues, in 7th International Conference on Computational Methods in Science and Engineering (ICCMSE), AIP Conference Proceedings, vol. 1504, pp. 364–376 (2009). doi:10.1063/1.4771729

  6. M. Ayvaz, M. Demiralp, A Fluctuation Analysis at the Classical Limit for the Expectation Dynamics of a Single Quartic Quantum Anharmonic Oscillator, in International Conference on Numerical Analysis and Applied Mathematics (ICNAAM), AIP Conference Proceedings, vol. 1281, pp. 1950–1953 (2010). doi:10.1063/1.3498311

  7. M. Ayvaz, M. Demiralp, Quantum Optimal Control of Single Harmonic Oscillator Under Quadratic Controls together with Linear Dipole Polarizability: A Fluctuation Free Expectation Value Dynamical Perspective, in International Conference on Numerical Analysis and Applied Mathematics (ICNAAM), AIP Conference Proceedings, vol. 1389, pp. 364–376 (2011) doi:10.1063/1.3637828

  8. M. Demiralp, E. Demiralp, L. Hernandez-Garcia, A probabilistic foundation for dynamical systems: theoretical background and mathematical formulation. J. Math. Chem. 58, 850–869 (2012)

    Article  Google Scholar 

  9. M. Demiralp, E. Demiralp, L. Hernandez-Garcia, A probabilistic foundation for dynamical systems: phenomenological reasoning and principal characteristics of probabilistic evolution. J. Math. Chem. 58, 870–880 (2012)

    Article  Google Scholar 

  10. M. Demiralp, E. Demiralp, A contemporary linear representation theory for ordinary differential equations: probabilistic evolutions and related approximants for unidimensional autonomous systems. J. Math. Chem. (2012). doi:10.1007/s10910-012-0070-2

    Google Scholar 

  11. M. Demiralp, E. Demiralp, A contemporary linear representation theory for ordinary differential equations: multilinear algebra in folded arrays (Folarrs) perspective and its use in multidimensional case. J. Math. Chem. (2012). doi:10.1007/s10910-012-0064-0

    Google Scholar 

  12. M. Demiralp, A probabilistic evolution approach trilogy, part 1: quantum expectation value evolutions, block triangularity and conicality, truncation approximants and their convergence, J. Math. Chem. (2012). doi:10.1007/s10910-012-0079-6

  13. M. Demiralp, N.A. Baykara, A probabilistic evolution approach tri logy, part 2: spectral issues for block triangular evolution matrix, singularities, space extension, J. Math. Chem. (2012). doi:10.1007/s10910-012-0080-0

  14. M. Demiralp, B. Tunga, A probabilistic evolution approach trilogy, part 3: temporal variation of state variable expectation values from Liouville equation perspective, J. Math. Chem. (2012) doi:10.1007/s10910-012-0081-z

  15. C. Gozukirmizi, M. Demiralp, Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations, part 1: arbitrariness and equipartition theorem in Kronecker power series, J. Math. Chem. 51(10) doi:10.1007/s10910-013-0298-5 (2013)

  16. C. Gozukirmizi, M. Demiralp, Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations, part 2: kernel separability, space extension, and, series solution via telescopic matrices, J. Math. Chem. 51(10) (2013). doi:10.1007/s10910-013-0299-4

  17. O.V. Prezhdo, Y.V. Pereverzev, Quantized Hamilton dynamics. J. Chem. Phys. 113(16), 6557–6565 (2000)

    Article  CAS  Google Scholar 

  18. O.V. Prezhdo, Y.V. Pereverzev, Quantized Hamilton dynamics for a general potential. J. Chem. Phys. 116(11), 4450–4461 (2002). doi:10.163/1.1451060

    Article  CAS  Google Scholar 

  19. Y. Shigeta, H. Miyachi, K. Hirao, Quantal cumulant dynamics: general theory. J. Chem. Phys. 125(24), 244102 (2006)

    Article  Google Scholar 

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Acknowledgments

The first author thanks to The Scientific and Technological Research Council of Turkey (TUBITAK) for their support during his PhD studies.

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  1. İstanbul Teknik Üniversitesi Bilişim Ensti tüsü, Maslak, 34469, Istanbul, Turkey

    Muzaffer Ayvaz & Metin Demiralp

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  1. Muzaffer Ayvaz
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  2. Metin Demiralp
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Correspondence to Muzaffer Ayvaz.

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Ayvaz, M., Demiralp, M. Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators, part II: the use of mathematical fluctuation theory. J Math Chem 52, 2294–2315 (2014). https://doi.org/10.1007/s10910-014-0381-6

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  • DOI: https://doi.org/10.1007/s10910-014-0381-6

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