WKB Analysis of Bohmian Dynamics

Figalli, Alessio; Klein, Christian; Markowich, Peter A.; Sparber, Christof

doi:10.1002/cpa.21487

Cited by 8 publications

(10 citation statements)

References 49 publications

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“…Schrodinger−Bohm Dynamics. Bohm dynamics 8,9 can be understood as a formal extension of the conventional Schroëdinger dynamics, 15…”

Section: Theorymentioning

confidence: 99%

“…Bohm dynamics , can be understood as a formal extension of the conventional Schröedinger dynamics, which supplies the time dependent configuration Q ( t ) of a closed (isolated) quantum system on the basis of its quantum pure state |Ψ( t )⟩ having a pilot role. For the quantum system with n Cartesian degrees of freedom, the notation Q ( t ) = ( Q 1 ( t ), Q 2 ( t ), ..., Q n ( t )) will be used for the instantaneous configuration, where Q k ( t ) is the coordinate of the k th degrees of freedom ( k = 1, 2, ..., n ).…”

Section: Theorymentioning

confidence: 99%

“…This equivalence has been extensively exploited for developing innovative algorithms to calculate the wave function dynamics 11,12 and to derive semiclassical approximations. 13,14 As Figalli and co-workers stated, 15 despite Bohm theory "as particle trajectories remains controversial from the physics point of view, its mathematical foundation is solid": the Bohm trajectory is well-defined for all times. 16,17 In our approach, we employ precisely a single Bohm trajectory, in opposition to the statistical ensemble, for identifying the nuclear positions.…”

Section: Introductionmentioning

confidence: 99%

“…In the original formulation by Bohm, , the equivalence with quantum mechanics is recovered by employing the ensemble of all possible configurations evolving along a swarm of trajectories that collectively reproduce the time dependence of the wave function. This equivalence has been extensively exploited for developing innovative algorithms to calculate the wave function dynamics , and to derive semiclassical approximations. , As Figalli and co-workers stated, despite Bohm theory “as particle trajectories remains controversial from the physics point of view, its mathematical foundation is solid”: the Bohm trajectory is well-defined for all times. , In our approach, we employ precisely a single Bohm trajectory, in opposition to the statistical ensemble, for identifying the nuclear positions.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quantum Stochastic Trajectories: The Fokker–Planck–Bohm Equation Driven by the Reduced Density Matrix

Avanzini

Moro

2018

J. Phys. Chem. A

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The quantum molecular trajectory is the deterministic trajectory, arising from the Bohm theory, that describes the instantaneous positions of the nuclei of molecules by assuring the agreement with the predictions of quantum mechanics. Therefore, it provides the suitable framework for representing the geometry and the motions of molecules without neglecting their quantum nature. However, the quantum molecular trajectory is extremely demanding from the computational point of view, and this strongly limits its applications. To overcome such a drawback, we derive a stochastic representation of the quantum molecular trajectory, through projection operator techniques, for the degrees of freedom of an open quantum system. The resulting Fokker-Planck operator is parametrically dependent upon the reduced density matrix of the open system. Because of the pilot role played by the reduced density matrix, this stochastic approach is able to represent accurately the main features of the open system motions both at equilibrium and out of equilibrium with the environment. To verify this procedure, the predictions of the stochastic and deterministic representation are compared for a model system of six interacting harmonic oscillators, where one oscillator is taken as the open quantum system of interest. The undeniable advantage of the stochastic approach is that of providing a simplified and self-contained representation of the dynamics of the open system coordinates. Furthermore, it can be employed to study the out of equilibrium dynamics and the relaxation of quantum molecular motions during photoinduced processes, like photoinduced conformational changes and proton transfers.

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“…Schrodinger−Bohm Dynamics. Bohm dynamics 8,9 can be understood as a formal extension of the conventional Schroëdinger dynamics, 15…”

Section: Theorymentioning

confidence: 99%

Section: Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum Stochastic Trajectories: The Fokker–Planck–Bohm Equation Driven by the Reduced Density Matrix

Avanzini

Moro

2018

J. Phys. Chem. A

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“…Right: The corresponding classical trajectories, exhibiting interference for ≥ * ≈ 0.2. For more details, see [1].…”

mentioning

confidence: 99%

AMS Fall Western & Central Sampler: Semiclassical Quantum Dynamics and Bohmian Trajectories

Sparber¹

2016

Notices Amer. Math. Soc.

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T he early days of the twentieth century saw the establishment of quantum mechanics as a novel description of our physical world. Ever since its invention, a basic problem concerns the connection between quantum mechanics and the older, well-understood theory of classical mechanics. It was accepted early on that classical mechanics should be understood as an emergent phenomenon of quantum mechanics, i.e., it should be recovered from the underlying quantum mechanical description when considered over special values of its physical parameters. When trying to follow this basic idea, one immediately faces an obstacle: quantum mechanics and classical mechanics are usually treated within two entirely different mathematical formalisms. While the former is based on the time-evolution of vectors in (infinite-dimensional) Hilbert spaces, the latter is concerned with the dynamics of point particles on a (finite-dimensional) phase space.Focussing on one of the simplest quantum mechanical systems, we consider the dynamics of a single quantum particle, say, an electron, under the influence of a static external force field mediated by a given real-valued potential ( ), where ∈ ℝ 3 denotes the spatial coordinate. The state of the electron at time ∈ ℝ is described by a wave function ( , ⋅) ∈ 2 (ℝ 3 ; ℂ) whose time evolution is governed by Erwin Schrödinger's fundamental equation from 1926:Here we have rescaled the original equation including all physical parameters (mass, charge, the Planck's constant ℏ, etc.) into dimensionless units such that only one (small) parameter > 0 remains. The latter plays the role of a

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Pilot-Wave Quantum Theory with a Single Bohm’s Trajectory

2015

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The representation of a quantum system as the spatial configuration of its constituents evolving in time as a trajectory under the action of the wave-function, is the main objective of the de Broglie–Bohm theory (or pilot wave theory). However, its standard formulation is referred to the statistical ensemble of its possible trajectories. The statistical ensemble is introduced in order to establish the exact correspondence (the Born’s rule) between the probability density on the spatial configurations and the quantum distribution, that is the squared modulus of the wave-function. In this work we explore the possibility of using the pilot wave theory at the level of a single Bohm’s trajectory, that is a single realization of the time dependent configuration which should be representative of a single realization of the quantum system. The pilot wave theory allows a formally self-consistent representation of quantum systems as a single Bohm’s trajectory, but in this case there is no room for the Born’s rule at least in its standard form. We will show that a correspondence exists between the statistical distribution of configurations along the single Bohm’s trajectory and the quantum distribution for a subsystem interacting with the environment in a multicomponent system. To this aim, we present the numerical results of the single Bohm’s trajectory description of the model system of six confined planar rotors with random interactions. We find a rather close correspondence between the coordinate distribution of one rotor, the others representing the environment, along its trajectory and the time averaged marginal quantum distribution for the same rotor. This might be considered as the counterpart of the standard Born’s rule when the pilot wave theory is applied at the level of single Bohm’s trajectory. Furthermore a strongly fluctuating behavior with a fast loss of correlation is found for the evolution of each rotor coordinate. This suggests that a Markov process might well approximate the evolution of the Bohm’s coordinate of a single rotor (the subsystem) and, under this condition, it is shown that the correspondence between coordinate distribution and quantum distribution of the rotor is exactly verified

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WKB Analysis of Bohmian Dynamics

Cited by 8 publications

References 49 publications

Quantum Stochastic Trajectories: The Fokker–Planck–Bohm Equation Driven by the Reduced Density Matrix

Quantum Stochastic Trajectories: The Fokker–Planck–Bohm Equation Driven by the Reduced Density Matrix

AMS Fall Western & Central Sampler: Semiclassical Quantum Dynamics and Bohmian Trajectories

Pilot-Wave Quantum Theory with a Single Bohm’s Trajectory

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