Bohmian Trajectories as Borders of Regions of Constant Probability

2023

J. Phys.: Conf. Ser.

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In Bohmian mechanics it is assumed that quantum objects are particles that follow real physical paths. These Bohmian trajectories are obtained by integrating the velocity field that occurs in the continuity equation. However, in the course of deriving these trajectories, the independent position variable x, appearing in the velocity field, is arbtirarily replaced by the time-dependent trajectory q(t). We show under which restrictions such a replacement can be justified, leading to a totally different interpretation of the Bohmian trajectories. They are not real paths of physical particles but borders of regions of constant probability, so-called quantiles, and therefore must be seen in the context of descriptive statistics. Thus they provide a tool complementary to the conventional statistical formulation of quantum mechanics. A possible extension to include open dissipative systems is also discussed, showing that the inclusion of an environment does not change our interpretation of the Bohmian trajectories.

“…These findings are in agreement with mathematically-motivated approaches by Oriols et al [8], Brandt et al [9] and Coffey et al [10]. For further details see [7].…”

Section: Consistent Derivation Of the Bohmian Trajectory And Physical...supporting

confidence: 90%

“…i.e., the dependent variable σ(V ) replaces the independent variable S. For further details, see [7]. How can this procedure be transferred to our Bohmian problem?…”

Section: Consistent Derivation Of the Bohmian Trajectory And Physical...mentioning

confidence: 99%

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On a Reformulation of Bohmian Mechanics Including Open Dissipative Systems

2023

J. Phys.: Conf. Ser.

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“…Madelung's formulation of wave mechanics was also later independently used by David Bohm in their deterministically inspired version of quantum mechanics [23,24] where he claimed the existence of real paths of (quantum) particles that can be obtained by integration of Equation ( 7). As we have shown recently [25], this deterministic viewpoint is incorrect and has to be replaced by a probabilistic one that differs from the usual probabilistic viewpoint taken in quantum mechanics. Nevertheless, Bohmian mechanics can still be helpful in the treatment of quantum systems, e.g., tunneling problems, particularly when performing numerical simulations [26,27].…”

Section: Conventional Quantum Hydrodynamicsmentioning

confidence: 98%

Diffusion Effect in Quantum Hydrodynamics

Bonilla

2022

Axioms

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In this paper, we introduce (at least formally) a diffusion effect that is based on an axiom postulated by Werner Heisenberg in the early days of quantum mechanics. His statement was that—in quantum mechanics—kinematical quantities such as velocity must be treated as complex matrices. In the hydrodynamic formulation of quantum mechanics according to Madelung, the complex Schrödinger equation is rewritten in terms of two real equations—a continuity equation and a modified Hamilton–Jacobi equation. Considering seriously Heisenberg’s axiom, the velocity occurring in the continuity equation should be replaced by a complex one, thus introducing a diffusion term with an imaginary diffusion coefficient. Therefore, in quantum mechanics, there should be a diffusion effect showing up in the dynamics. This is the case in the time evolution of the free-motion wave packet under time reversal. The maximum returns to the initial position; however, the width of the wave packet does not shrink to its initial width. This effect is obvious but—as far as we know—it is not mentioned in any textbook. In our paper, we discuss this effect in detail and show the connection with a complex version of quantum hydrodynamics.

“…Shortly afterwards, in 1952, Bohm [ 14 , 15 ] independently took up Madelung’s idea, but tried to give it a deterministic interpretation, assuming that the integration of Madelung’s velocity

would provide trajectories that are real paths followed by physical particles. (For a comprehensive review of the criticism of Bohm’s interpretation see [ 16 , 17 , 18 , 19 ] and the references cited therein; a consistent explanation of the Bohmian trajectories in probabilistic terms was given recently in [ 20 ].) Despite the ontological problems, the application of the Bohmian approach developed very useful numerical methods for the treatment of molecular reactions, tunnelling, and diffraction phenomena (see [ 21 ] for a review).…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Relations in the Madelung Picture

2021

Entropy

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Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.