Jump-type processes and their applications in quantum mechanics

Maslov, V. P.; Chebotarev, A. M.

doi:10.1007/bf01088985

Cited by 8 publications

(4 citation statements)

References 51 publications

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“…Let A be a constant magnetic potential and V a sum of |x| 2 and the Fourier transform of a complex measure of bounded variation on R n . The path integral for such potentials was studied in [1,2,12,13,15,19] and so on in various ways. The path integral formulated through piecewise classical paths was studied by Fujiwara and Yajima in [7,8,20] for a wider class of potentials than the above.…”

Section: Introductionmentioning

confidence: 99%

On Convergence of the Feynman Path Integral Formulated Through Broken Line Paths

Ichinose

1999

Rev. Math. Phys.

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We study the convergence of the Feynman path integral formulated through broken line paths in nonrelativistic quantum mechanics. The rigorous proof of its convergence had been given little except for special cases for a long time. In the preceding paper the author showed for a class of potentials that this path integral converges and gives the probability amplitude, i.e. the solution of the Schrödinger equation. In the present paper we generalize this result, showing the boundedness theorem on the weighted Sobolev spaces for some integral operators and joining this boundedness theorem to the method in the preceding paper. We note that the result obtained is gauge invariant.

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Section: Introductionmentioning

confidence: 99%

On Convergence of the Feynman Path Integral Formulated Through Broken Line Paths

Ichinose

1999

Rev. Math. Phys.

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“…To obtain a numerical method that is directly applicable to truly multidimensional systems, it would be desirable to employ a stochastic scheme involving a Monte Carlo sampling of local classical trajectories. Based on a stochastic description of quantummechanical time evolution, 39,40 several works have addressed the closely related problem of a Monte Carlo implementation of the quantum-mechanical Wigner-Moyal equation. [41][42][43] Such an approach has to cope with two major complications, that is, the representation of nonlocal phase-space operators and the convergence of the sampling procedure which is cumbersome due to complex-valued trajectories with rapidly oscillating phases.…”

Section: ͑13͒mentioning

confidence: 99%

“…While this scheme is straightforward to evaluate using a grid in phase space, 28 its implementation on a grid becomes prohibitive in the case of multidimensional systems. Following previous work, [39][40][41][42][43] we therefore introduce a stochastic algorithm which uses a Monte Carlo sampling of local classical trajectories. To this end, the density matrix is represented as an ensemble of phase-space points described by the weight functions W nm (x,p,t), the phase functions nm (x,p,t), and the density P nm (x,p,t),…”

Section: ͑32͒mentioning

confidence: 99%

Quantum-classical Liouville description of multidimensional nonadiabatic molecular dynamics

Santer

Manthe

Stock

2001

The Journal of Chemical Physics

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The quantum-classical Liouville formulation gives a quantum-mechanical density-matrix description of the ''quantum'' particles of a problem ͑e.g., the electrons͒ and a classical phase-space-density description of the ''classical'' particles ͑e.g., the nuclei͒. In order to employ this formulation to describe multidimensional nonadiabatic processes in complex molecular systems, this work is concerned with an efficient Monte Carlo implementation of the quantum-classical Liouville equation. Although an exact stochastic realization of this equation is in principle available, in practice one has to cope with two major complications: ͑i͒ The representation of nonlocal phase-space operators in terms of local classical trajectories and ͑ii͒ the convergence of the Monte Carlo sampling which is cumbersome due to complex-valued trajectories with rapidly oscillating phases. Several strategies to cope with these problems are discussed, including various approximations to determine the momentum shift associated with a nonadiabatic transition, the on-the-fly generation of new trajectories at curve-crossings, and the localization of trajectories after irreversible electronic transitions. Employing several multidimensional model systems describing ultrafast photoinduced electron transfer and internal conversion, detailed numerical studies are performed which are compared to exact quantum calculations as well as to the ''fewest-switches'' surface-hopping method. In all cases under consideration, the Liouville calculations are in good agreement with the quantum reference. In particular, the approach is shown to provide a correct quantum-classical description of the electronic coherence.

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“…The studies of Kac [3] and Ito [4,5] gave the first mathematically rigorous results concerning integral representations of the solutions of analogous partial differential equations. Sub-sequently~ these types of formulas were considered in a large number of studies devoted to the application of mathematical methods of path integration in the investigation of equations [6,7] and the bibliography contained therein).…”

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confidence: 99%

Feynman's formula in phase space for systems of pseudodifferential equations with analytic symbols

Egikyan

Ktitarev

1992

Math Notes

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Jump-type processes and their applications in quantum mechanics

Cited by 8 publications

References 51 publications

On Convergence of the Feynman Path Integral Formulated Through Broken Line Paths

On Convergence of the Feynman Path Integral Formulated Through Broken Line Paths

Quantum-classical Liouville description of multidimensional nonadiabatic molecular dynamics

Feynman's formula in phase space for systems of pseudodifferential equations with analytic symbols

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