Flux continuity and probability conservation in complexified Bohmian mechanics

Poirier, Bill

doi:10.1103/physreva.77.022114

Cited by 41 publications

(39 citation statements)

References 44 publications

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“…To conclude, we summarize the positive and negative features of this possible definition of quantum probability density in the complex space, in comparison with the distribution prescribed in [18]. First, we see that our definition helps to obtain the Born's Ψ ⋆ Ψ probability density along the real line from the velocity field in the complex trajectory formalism, in a nontrivial way.…”

Section: Discussionmentioning

confidence: 99%

“…Ref. [18] strongly advocates an analytic expression for ρ, in view of its anticipated advantages in the synthetic time-dependent Schrodinger equation applications. But the definition in [18] is shown not to lead to a continuity equation and it is presented there as a negative result.…”

Section: Discussionmentioning

confidence: 99%

“…[18] strongly advocates an analytic expression for ρ, in view of its anticipated advantages in the synthetic time-dependent Schrodinger equation applications. But the definition in [18] is shown not to lead to a continuity equation and it is presented there as a negative result. Whether our definition of extended probability in this paper, which does obey a continuity equation, is helpful for synthetic applications is an open problem, and is worth pursuing.…”

Section: Discussionmentioning

confidence: 99%

“…But in this case, it becomes necessary to see whether probability conservation holds everywhere in the plane. A recent paper by Poirier [18] addresses this issue and obtains some negative results for the choices made for such a distribution. Poirier tries to define an extended probability by working with Ψ, the solution of time-dependent Schrodinger equation.…”

Section: Introductionmentioning

confidence: 99%

“…This, arguably leads to nonconservation of probability along trajectories. But it shall be reminded that this negative result is based on the choices made in [18] for the probability density and flux.…”

Section: Introductionmentioning

confidence: 99%

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Probability and complex quantum trajectories

John¹

2009

Annals of Physics

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It is shown that in the complex trajectory representation of quantum mechanics, the Born's Ψ ⋆ Ψ probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Probability and complex quantum trajectories

John¹

2009

Annals of Physics

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Computational investigation of wave packet scattering in the complex plane: Numerical analytic continuation techniques

Rowland

Wyatt

2010

Int J of Quantum Chemistry

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ABSTRACT:In this study, we utilize Numerical Analytic Continuation (NAC) techniques to compute the complex-extension of wave packets computed on the realaxis. Various methods for doing this are reviewed, and the accuracy and stability of one particular method (the Cauchy Method) are examined using a test function. We focus on generating numerically accurate maps of the complex-extended time-dependent wave packet for the Gaussian and Eckart barrier scattering problems. We also examine quantum momentum function vector fields and the associated Pó lya vector fields for this problem. In a companion study, we present a novel method for solving the complex-extended Schrö dinger equation for points in the complex plane for wave packet scattering from a Gaussian barrier. Several fascinating features of the solution (such as quantized vortices) were presented, but we could only check the veracity of those results on the real-axis. The method utilized in this study provides an additional check of these results.

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Electronic quantum motions in hydrogen molecule ion

Yang

Weng

2011

Int J of Quantum Chemistry

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ABSTRACT:The hydrogen molecule ion is a two-center force system expressed under the prolate spheroidal coordinates, whose quantum motions and quantum trajectories have never been addressed in the literature before. The momentum operators in this coordinate system are derived for the first time from the Hamilton equations of motion and used to construct the Hamiltonian operator. The resulting Hamiltonian comprises a kinetic energy T and a total potential V Total consisting of the Coulomb potential and a quantum potential. It is shown that the participation of the quantum potential and the accompanied quantum forces in the force interaction within H 2 þ is essential to develop an electronic motion consistent with the prediction of the probability density function |W| 2 . The motion of the electron in H 2 þ can be either described by the Hamilton equations derived from the Hamiltonian H ¼ T K þ V Total or by the Lagrange equations derived from the Lagrangian H ¼ T K À V Total . Solving the equations of motion with different initial positions, we show that the solutions yield an assembly of electronic quantum trajectories whose distribution and concentration reconstruct the r and p molecular orbitals in H 2 þ .

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Flux continuity and probability conservation in complexified Bohmian mechanics

Cited by 41 publications

References 44 publications

Probability and complex quantum trajectories

Probability and complex quantum trajectories

Computational investigation of wave packet scattering in the complex plane: Numerical analytic continuation techniques

Electronic quantum motions in hydrogen molecule ion

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