One scheme of Transition Probability Evaluation

In present work are considered the scheme of evaluation the transition probability in quantum system. It is based on path integral representation of transition probability amplitude and its evaluation by means of a saddle point method, applied to the part of integration variables. The whole integration process is reduced to initial value problem solutions of Hamilton equations with a random initial phase point. The scheme is related to the semiclassical initial value representation approaches using great number of trajectories. In contrast to them from total set of generated phase paths only one path for each initial coordinate value is selected in Monte Karlo process.

Authors:

• Alexander B. Bichkov
• Alla A. Mityureva
• Valery V. Smirnov

Keywords:

• Path integral
• saddle point method
• semiclassical approximation
• transition probability

References:

[1] J. R. Klauder. Path integrals and stationary-phase approximations. Phys.
Rev. D 19, p. 2349-2356 (1979).
[2] V. V. Smirnov. On the estimation of a path integral by means of the
saddle point method. J. Phys. A: Math. and Theor., 2010, v. 43, 465303
(11pp)
[3] M. Baranger, M. A. M. de Aguiar, F. Keck, H. J. Korsch and B.
Schellhaas. Semiclassical approximations in phase space with coherent
states. J. Phys. A: Math. Gen., 2001, v. 34, p. 7227-7286
[4] J. H. Samson. Phase-space path-integral calculation of the Wigner
function. J. Phys. A: Math. Gen., 2003, v. 36, p. 10637-10650
[5] Jeremy Schiff, Yair Goldfarb and David J. Tannor. Path integral
derivations of novel complex trajectory methods. arXiv:0807.4659v2
[quant-ph] 30 Jul 2008
[6] M. S. Marinov. Path integrals on homogeneous manifolds. J. Math.
Phys., 1995, v. 36, n. 5, p. 2458-2469
[7] Carol Braun and Anupam Garg. Semiclassical coherent-state propagator
for many particles. J. Math. Phys., 2007, v. 48, 032104
[8] Carol Braun and Anupam Garg. Semiclassical coherent-state propagator
for many spins. J. Math. Phys., 2007, v. 48, 102104
[9] V. N. Kolokoltsov. Complex calculus of variations, infinite-dimensional
saddle-point method and Feynman integral for dissipative stochastic
Schroedinger equation. Preprint, Nottingham Trent University, 1999.
[10] Hagen Kleinert. Path integrals in quantum mechanics, statistics,
polymer physics, and financial markets. (World Scientific Publishing
Co. 5th Edition, 2010)
[11] M. F. Herman and E. Kluk. A semiclassical justification for the use of
non-spreading wavepackets in dynamics calculations. Chem.
Phys.1984, v. 91, p. 27-34
[12] M. F. Herman. Dynamics by semiclassical methods Annu. Rev. Phys.
Chem. 1994, v. 45, p. 83-111
[13] Sarin A Deshpande and Gregory S Ezra. On the derivation of the
Herman-Kluk propagator. J. Phys. A: Math. Gen. 39 (2006) 5067-5078
[14] Eric J. Heller. Cellular dynamics: A new semiclassical approach to timedependent
quantum mechanics. J. Chem. Phys. 94 (4), 1991, p. 2723-
2729
[15] Kenneth G. Kay. Integral expressions for the semiclassical timedependent
propagator. J. Chem. Phys. 100 (6), 1994, p. 4377-4392
[16] Frank Grossmann and Michael F. Herman. Comment on ÔÇÿSemiclassical
approximations in phase space with coherent states-. J. Phys. A: Math.
Gen. 35 (2002) 9489-9492
[17] C. Harabatia and J. M. Rost, F. Grossmann. Long-time and unitary
properties of semiclassical initial value representations. Journal of
Chemical Physics, 2004, v. 120, n. 1 p. 26-30
[18] A. M. Perelomov. Generalized coherent states and their applications.
(Springer-Verlag, 1986)
[19] F. A. Berezin. Secondary quantization method. (Moscow, Nauka, 1986)
[20] J. R. Klauder and B. S. Skagerstam. Coherent States: Applications in
Physics and Mathematical Physics. (Singapore: World Scientific, 1985)
[21] V. V. Smirnov. A note on the limiting procedures for path integrals. J.
Phys. A: Math. and Theor., 2008, v. 41, 035306
[22] L. C. dos Santos and M. A. M. de Aguiar. Coherent state path integrals
in the Weyl representation. J. Phys. A: Math. Gen. 39 (2006) 13465-
13482
[23] J. R. Klauder. Path integrals and stationary-phase approximations. Phys.
Rev. D 19, p. 2349-2356 (1979).
[24] V. V. Smirnov. Path integral for system with spin. J. Phys. A : Math.
and Gen., 1999, v.32, n.7, p.1285-1290
[25] V. V. Smirnov and A. A. Mityureva. The new approach to the computer
experimental studies of atoms excitation by electron impact. Helium. J.
Phys. B. : Atomic and Mol. Phys., 1996, v.29, n.13, p.2865-2874
[26] A. B. Bichkov, A. A. Mityureva and V. V. Smirnov. Path-integral-based
evaluation of the probability of hydrogen atom ionization by short
photo-pulse. J. Phys. B: At. Mol. Opt. Phys., 2011, v. 44, 135601 (6pp)
[27] M. A. Evgrafov. Asymptotic estimates and entire functions. (New York,
Gordon & Breach, 1962)