Computational method for the quantum Hamilton-Jacobi equation: One-dimensional scattering problems

Chou, Chia‐Chun; Wyatt, Róbert E.

doi:10.1103/physreve.74.066702

Cited by 39 publications

(31 citation statements)

References 30 publications

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“…In other words, this complexified Bohmian mechanics is also an alternative synthetic method but in complex space [193]. With the same spirit and by the same time, an alternative synthetic approach was also developed by Chou and Wyatt [407][408][409], with applications to both bound states and scattering systems.…”

Section: Trajectories From Complex Actionmentioning

confidence: 99%

Applied Bohmian mechanics

et al. 2014

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Abstract. Bohmian mechanics provides an explanation of quantum phenomena in terms of point-like particles guided by wave functions. This review focuses on the use of nonrelativistic Bohmian mechanics to address practical problems, rather than on its interpretation. Although the Bohmian and standard quantum theories have different formalisms, both give exactly the same predictions for all phenomena. Fifteen years ago, the quantum chemistry community began to study the practical usefulness of Bohmian mechanics. Since then, the scientific community has mainly applied it to study the (unitary) evolution of single-particle wave functions, either by developing efficient quantum trajectory algorithms or by providing a trajectorybased explanation of complicated quantum phenomena. Here we present a large list of examples showing how the Bohmian formalism provides a useful solution in different forefront research fields for this kind of problems (where the Bohmian and the quantum hydrodynamic formalisms coincide). In addition, this work also emphasizes that the Bohmian formalism can be a useful tool in other types of (nonunitary and nonlinear) quantum problems where the influence of the environment or the nonsimulated degrees of freedom are relevant. This review contains also examples on the use of the Bohmian formalism for the many-body problem, decoherence and measurement processes. The ability of the Bohmian formalism to analyze this last type of problems for (open) quantum systems remains mainly unexplored by the scientific community. The authors of this review are convinced that the final status of the Bohmian theory among the scientific community will be greatly influenced by its potential success in those types of problems that present nonunitary and/or nonlinear quantum evolutions. A brief introduction of the Bohmian formalism and some of its extensions are presented in the last part of this review.

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Section: Trajectories From Complex Actionmentioning

confidence: 99%

Applied Bohmian mechanics

et al. 2014

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“…As an example, complex trajectories in the first excited state of harmonic oscillator is shown in figures 1. This formalism was extended to three dimensional problems such as the hydrogen atom by Yang [14] and was used to investigate one dimensional scattering problems and bound state problems by Chou and Wyatt [15]. Later, a complex trajectory approach for solving the QHJE was developed by Tannor and co-workers [16].…”

Section: Complex Quantum Trajectoriesmentioning

confidence: 99%

“…which is equivalent to (15). For the RW spacetime which contains only a cosmological constant, (17) helps us to write the constraint equation as…”

Section: Hamiltonian Formulationmentioning

confidence: 99%

Exact classical correspondence in quantum cosmology

John¹

2015

Gravit. Cosmol.

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We find a Friedmann model with appropriate matter/energy density such that the solution of the Wheeler-DeWitt equation exactly corresponds to the classical evolution. The well-known problems in quantum cosmology disappear in the resulting coasting evolution. The exact quantum-classical correspondence is demonstrated with the help of the de Broglie-Bohm and modified de Broglie-Bohm approaches to quantum mechanics. It is reassuring that such a solution leads to a robust model for the universe, which agrees well with cosmological expansion indicated by SNe Ia data.

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“…The resultant "complex-valued Bohmian trajectories" offer certain advantages; for instance, they are known not to be fixed-points, in general, for nondegenerate stationary states, so that it is possible to achieve nontrivial trajectory dynamics in this context. Although complexvalued Bohmian mechanics may still be in its infancy, interest has grown tremendously in the last few years [1,9,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. The field appears to have started in the 1980's with a paper by Leacock and Padgett [15] and a less well known (and very brief) article by Tourenne [16].…”

Section: Introductionmentioning

confidence: 99%

Flux continuity and probability conservation in complexified Bohmian mechanics

Poirier

2008

Phys. Rev. A

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Recent years have seen increased interest in complexified Bohmian mechanical trajectory calculations for quantum systems, both as a pedagogical and computational tool. In the latter context, it is essential that trajectories satisfy probability conservation, to ensure they are always guided to where they are most needed. In this paper, probability conservation for complexified Bohmian trajectories is considered. The analysis relies on time-reversal symmetry considerations, leading to a generalized expression for the conjugation of wavefunctions of complexified variables. This in turn enables meaningful discussion of complexified flux continuity, which turns out not to be satisfied in general, though a related property is found to be true. The main conclusion, though, is that even under a weak interpretation, probability is not conserved along complex Bohmian trajectories.

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Computational method for the quantum Hamilton-Jacobi equation: One-dimensional scattering problems

Cited by 39 publications

References 30 publications

Applied Bohmian mechanics

Applied Bohmian mechanics

Exact classical correspondence in quantum cosmology

Flux continuity and probability conservation in complexified Bohmian mechanics

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