On the classical limit of Bohmian mechanics for Hagedorn wave packets

Dürr, Detlef; Römer, Sarah

doi:10.1016/j.jfa.2010.07.011

Cited by 11 publications

(11 citation statements)

References 13 publications

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“…Semiclassical wave packets. In [33] a result similar to Theorem 3.1 (ii) is proved, for the case of semiclassical wave-packets (see also [16] for a closely related study). The corresponding initial data are of the form (6.2) ψ ε 0 (x) = ε −d/4 a 0…”

Section: Numerical Simulation Of Bohmian Trajectoriesmentioning

confidence: 74%

WKB analysis of Bohmian dynamics

Figalli¹,

Klein²,

Markowich³

et al. 2012

Preprint

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We consider a semi-classically scaled Schrödinger equation with WKB initial data. We prove that in the classical limit the corresponding Bohmian trajectories converge (locally in measure) to the classical trajectories before the appearance of the first caustic. In a second step we show that after caustic onset this convergence in general no longer holds. In addition, we provide numerical simulations of the Bohmian trajectories in the semiclassical regime which illustrate the above results.

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Section: Numerical Simulation Of Bohmian Trajectoriesmentioning

confidence: 74%

WKB analysis of Bohmian dynamics

Figalli¹,

Klein²,

Markowich³

et al. 2012

Preprint

View full text Add to dashboard Cite

show abstract

“…One should note that the mathematical methods used in [9] are rather different from ours and that no convergence result for P ε (t) is given, except for p 0 = 0 (and, as a variant, for a class of time-averaged Bohmian momenta). In comparison to that, the use of β ε (t), together with classical Young measure theory, allows us to conclude the following result:…”

Section: Resultsmentioning

confidence: 91%

“…Finally, we shall consider the particular case where the initial data ψ ε 0 is a socalled semi-classical wave packet, see Theorem 2.3 below. The classical limit of Bohmian trajectories in this particular situation has been recently studied in [9]. There it has been shown that the Bohmian trajectories X ε (t, x) converge (in some suitable topology) to X(t), the classical particle trajectory induced by the Hamiltonian system…”

Section: Resultsmentioning

confidence: 99%

On the Dynamics of Bohmian Measures

Markowich

Paul

Sparber

2012

Arch Rational Mech Anal

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Abstract. The present work is devoted to the study of dynamical features of Bohmian measures, recently introduced by the authors. We rigorously prove that for sufficiently smooth wave functions the corresponding Bohmian measure furnishes a distributional solution of a nonlinear Vlasov-type equation. Moreover, we study the associated defect measures appearing in the classical limit. In one space dimension, this yields a new connection between monokinetic Wigner and Bohmian measures. In addition, we shall study the dynamics of Bohmian measures associated to so-called semi-classical wave packets. For these type of wave functions, we prove local in-measure convergence of a rescaled sequence of Bohmian trajectories towards the classical Hamiltonian flow on phase space. Finally, we construct an example of wave functions whose limiting Bohmian measure is not mono-kinetic but nevertheless equals the associated Wigner measure.

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“…0 C . Indeed, it has been proved in [44] that for the case of semiclassical wave packets, convergence of the Bohmian flow to its corresponding classical counterpart holds in some appropriate topology (see also [21] for a closely The main differences between WKB states and semiclassical wave packets are that for the latter, the particle density concentrates in a point, i.e., " 0 .x/ "!0 C ! ı.x x 0 / in D 0 .R d /, and that the corresponding classical phase space flow does not exhibit caustics; cf.…”

Section: Semiclassical Wave Packetsmentioning

confidence: 99%

WKB Analysis of Bohmian Dynamics

Figalli

Klein

Markowich

et al. 2013

Comm Pure Appl Math

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We consider a semiclassically scaled Schrödinger equation with WKB initial data. We prove that in the classical limit the corresponding Bohmian trajectories converge (locally in measure) to the classical trajectories before the appearance of the first caustic. In a second step we show that after caustic onset this convergence in general no longer holds. In addition, we provide numerical simulations of the Bohmian trajectories in the semiclassical regime that illustrate the above results.© 2014 Wiley Periodicals, Inc.

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On the classical limit of Bohmian mechanics for Hagedorn wave packets

Abstract: We consider the classical limit of quantum mechanics in terms of Bohmian trajectories. For wave packets as defined by Hagedorn we show that the Bohmian trajectories converge to Newtonian trajectories in probability.Comment: some minor changes; published versio

Cited by 11 publications

References 13 publications

WKB analysis of Bohmian dynamics

WKB analysis of Bohmian dynamics

On the Dynamics of Bohmian Measures

WKB Analysis of Bohmian Dynamics

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