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

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• 作者：Muzaffer Ayvaz (1)
  Metin Demiralp (1)
  
• 关键词：Probabilistic evolution approach ; Quantum mechanics ; Ehrenfest Theorem ; Expectation value dynamics ; Kronecker power series ; Mathematical fluctuation theory ; Quantized Hamilton dynamics ; Quantal cumulant dynamics ; 15A18 ; 34A05 ; 34A12
• 刊名：Journal of Mathematical Chemistry
• 出版年：2014
• 出版时间：September 2014
• 年：2014
• 卷：52
• 期：8
• 页码：2294-2315
• 全文大小：567 KB
• 参考文献：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
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  4. B. Tunga, M. Demiralp, Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation. J. Math. Chem. 49, 894-09 (2012). doi:10.1007/s10910-010-9786-z CrossRef
  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. 364-76 (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. 1950-953 (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. 364-76 (2011) doi:10.1063/1.3637828
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  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
  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
  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
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  18. O.V. Prezhdo, Y.V. Pereverzev, Quantized Hamilton dynamics for a general potential. J. Chem. Phys. 116(11), 4450-461 (2002). doi:10.163/1.1451060 CrossRef
  19. Y. Shigeta, H. Miyachi, K. Hirao, Quantal cumulant dynamics: general theory. J. Chem. Phys. 125(24), 244102 (2006) CrossRef
  
• 作者单位：Muzaffer Ayvaz (1)
  Metin Demiralp (1)
  
  1. ?stanbul Teknik üniversitesi Bili?im Ensti tüsü, Maslak, 34469, Istanbul, Turkey
  
• ISSN：1572-8897

文摘

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.