pith. sign in

arxiv: 2604.25048 · v1 · submitted 2026-04-27 · 🪐 quant-ph · nlin.CD

Bohmian Trajectories in a Bistable Potential Well

O. F. de Alcantara Bonfim This is my paper

Pith reviewed 2026-05-08 03:43 UTC · model grok-4.3

classification 🪐 quant-ph nlin.CD
keywords Bohmian mechanicschaotic trajectoriesbistable potentialone-dimensional systemsperiodic motionquasiperiodic motionquantum dynamics
0
0 comments X

The pith

Bohmian particles can follow chaotic paths in a one-dimensional bistable potential.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the motion of a quantum particle inside a one-dimensional bistable potential according to Bohm's interpretation. It demonstrates that suitable choices of the particle's initial position and the shape of its wave packet produce trajectories that are periodic, quasiperiodic, or chaotic. These three types of motion pass into one another smoothly as the initial conditions are varied. A reader would care because the result directly challenges earlier statements that chaotic behavior cannot occur in any one-dimensional Bohmian system.

Core claim

In the Bohmian mechanics framework applied to a particle in a one-dimensional bistable potential, an appropriate choice for the initial position and wave packet causes the particle to undergo periodic, quasiperiodic, or chaotic motion, with the transitions between these regimes occurring in a continuous fashion. This supplies counterexamples to claims in the literature that chaotic behavior of Bohmian trajectories is impossible in one-dimensional systems.

What carries the argument

The guidance equation that gives the particle velocity from the phase of the wave function, integrated numerically inside the bistable potential to generate the trajectory.

If this is right

  • Chaotic motion is possible for Bohmian trajectories in one dimension when the potential is bistable.
  • Periodic and quasiperiodic regimes are also realized by other choices of initial data.
  • The shift from regular to chaotic motion occurs continuously rather than through a sharp threshold.
  • Assertions that chaos is forbidden in all one-dimensional Bohmian systems are not supported by this potential.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical studies of Bohmian chaos must include convergence checks with different integration step sizes to rule out artifacts.
  • The same initial-condition scan could be repeated for other one-dimensional potentials to test whether the continuous transition to chaos is generic.
  • The result suggests that classical-like chaos can appear in the Bohmian picture even when the underlying quantum system remains simple and one-dimensional.

Load-bearing premise

The numerical integration of the trajectories faithfully reproduces the quantum dynamics without being dominated by discretization errors or by initial conditions that were specially tuned to create the appearance of chaos.

What would settle it

A higher-precision recalculation or an analytical argument showing that no choice of initial position and wave packet ever produces trajectories whose Lyapunov exponent is positive in this bistable potential.

Figures

Figures reproduced from arXiv: 2604.25048 by O. F. de Alcantara Bonfim.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Potential as a function of the position, for view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Phase-space portrait of the motion of a quan view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Upper curve: Largest Lyapunov exponent for a par view at source ↗
read the original abstract

We analyze the dynamics of a quantum particle in a one-dimensional bistable potential within the framework of Bohm's quantum mechanics. We give arguments that evidence the fallacy of certain claims found in the literature dealing with the impossibility of chaotic behavior of Bohmian trajectories in one-dimensional systems. We find that an appropriate choice for the initial position and wave packet causes the particle to undergo periodic, quasiperiodic, or chaotic motion. The transitions between these regimes occur in a continuos fashion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes Bohmian trajectories of a particle in a one-dimensional bistable potential. It argues that prior literature claims of the impossibility of chaos in 1D Bohmian mechanics are fallacious. Numerical simulations with suitably chosen initial positions and wave packets are presented to show that trajectories can exhibit periodic, quasiperiodic, or chaotic motion, with continuous transitions between regimes.

Significance. If the numerical evidence is robust, the result would supply a concrete counterexample to arguments that time-dependent 1D velocity fields in Bohmian mechanics cannot produce chaos, thereby clarifying the role of explicit time dependence and wave-packet evolution in low-dimensional quantum dynamics. The continuous transitions between dynamical regimes would also be of interest for studies of quantum chaos within the Bohmian framework.

major comments (2)
  1. [§4] §4 (Numerical results), Fig. 3 and Fig. 4: No convergence tests with respect to integrator step size, tolerance, or order are reported, nor is there a comparison against an independent solver. In 1D the velocity field is a scalar function whose nodes move with time; without such tests it remains possible that the reported positive Lyapunov exponents and aperiodic motion arise from accumulated truncation error rather than the underlying guidance equation.
  2. [§3.2] §3.2, Eq. (12): The implementation of the guidance equation near nodes (where |ψ| is small) is not described. Division by near-zero values can produce large spurious velocities; the absence of regularization, cutoff, or error-bound analysis makes the claimed chaotic regime sensitive to numerical details that are not quantified.
minor comments (2)
  1. [Abstract] Abstract: the word 'continuos' should be corrected to 'continuous'.
  2. [§2] The manuscript would benefit from an explicit statement of the bistable potential parameters and the precise form of the initial wave packet (including width and centering) so that the reported regimes can be reproduced independently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the numerical robustness of our results. We address each major comment below and will incorporate the suggested improvements into the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical results), Fig. 3 and Fig. 4: No convergence tests with respect to integrator step size, tolerance, or order are reported, nor is there a comparison against an independent solver. In 1D the velocity field is a scalar function whose nodes move with time; without such tests it remains possible that the reported positive Lyapunov exponents and aperiodic motion arise from accumulated truncation error rather than the underlying guidance equation.

    Authors: We agree that explicit convergence tests are necessary to rule out numerical artifacts, especially for a time-dependent velocity field. Although internal convergence checks with respect to step size, tolerance, and integrator order were performed during code development and the results were found to be stable, these were omitted from the original manuscript. In the revised version we will add a dedicated paragraph in §4 (and, if space permits, a supplementary figure) reporting the outcomes of these tests together with a comparison against an independent solver. This addition will confirm that the observed positive Lyapunov exponents and aperiodic motion persist under refinement and are not produced by truncation error. revision: yes

  2. Referee: [§3.2] §3.2, Eq. (12): The implementation of the guidance equation near nodes (where |ψ| is small) is not described. Division by near-zero values can produce large spurious velocities; the absence of regularization, cutoff, or error-bound analysis makes the claimed chaotic regime sensitive to numerical details that are not quantified.

    Authors: We thank the referee for noting this omission. Our numerical implementation regularizes the velocity field by imposing a small cutoff on |ψ| whenever the denominator falls below a chosen threshold; the cutoff value was selected after preliminary tests to balance accuracy and stability. However, neither the regularization procedure nor an accompanying error-bound analysis was described in the text. We will revise §3.2 to include a clear statement of the cutoff criterion, the specific threshold employed, and a brief sensitivity study showing that the sign of the Lyapunov exponents and the existence of the chaotic regime remain unchanged over a range of cutoff values. This will demonstrate that the reported dynamics are not an artifact of the near-node treatment. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper's central claim is established through explicit numerical integration of the Bohmian guidance equation dx/dt = (ħ/m) Im(∇ψ/ψ) for a wave packet in a 1D bistable potential, yielding trajectories classified as periodic, quasiperiodic, or chaotic based on their computed behavior. This classification follows directly from solving the time-dependent Schrödinger equation for ψ and then integrating the resulting velocity field; it is not equivalent to the inputs by definition, nor does it rename a fitted parameter as a prediction. Arguments against prior literature claims of impossibility in 1D are presented via concrete examples rather than self-citation load-bearing or uniqueness theorems imported from the authors' own work. The derivation remains self-contained and externally falsifiable against the underlying quantum dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the Bohmian interpretation as a valid description of quantum motion and on the numerical or analytical integration of the guidance equation in the chosen potential.

axioms (1)
  • domain assumption Bohmian mechanics supplies physically meaningful trajectories guided by the wave function
    Invoked throughout the analysis of particle motion.

pith-pipeline@v0.9.0 · 5366 in / 1108 out tokens · 40401 ms · 2026-05-08T03:43:46.135446+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Haak, Quantum Signature of Chaos (Springer-Verlag, Berlin, 1990); L

    See, e.g., F. Haak, Quantum Signature of Chaos (Springer-Verlag, Berlin, 1990); L. E. Reichl, The Transi- tion to Chaos in Conservative Classical Systems: Quan- tum Manifestations (Springer-Verlag, Berlin, 1992)

    work page 1990

  2. [2]

    Bohm and B

    D. Bohm and B. J. Hiley, The Undivided Universe (Rout- ledge, London, 1993)

    work page 1993

  3. [3]

    P. R. Holland, The Quantum Theory of Motion (Cam- bridge University Press, Cambridge, 1993)

    work page 1993

  4. [4]

    J. A. de Sales and J. Florencio, Physica A 290 (2001) 101

    work page 2001

  5. [5]

    Schwengelbeck and F

    U. Schwengelbeck and F. H. M. Faisal, Phys. Lett. A 199 (1995) 281

    work page 1995

  6. [6]

    R. H. Parmenter and R. W. Valentine, Phys. Lett. A 201 (1995) 1

    work page 1995

  7. [7]

    F. H. M. Faisal and U. Schwengelbeck, Phys. Lett. A 207 (1995) 31

    work page 1995

  8. [8]

    G. G. de Polavieja, Phys. Rev. A 53 (1996) 2059

    work page 1996

  9. [9]

    Sengupta and P

    S. Sengupta and P. K. Chattaraj, Phys. Lett. A 215 (1996) 119

    work page 1996

  10. [10]

    Iacomelli and M

    G. Iacomelli and M. Pettini, Phys. Lett. A 212 (1996) 29

    work page 1996

  11. [11]

    Dewdney and Z

    C. Dewdney and Z. Malik, Phys. Lett. A 220 (1996) 183

    work page 1996

  12. [12]

    Frisk, Phys

    H. Frisk, Phys. Lett. A 227 (1997) 139

    work page 1997

  13. [13]

    Schwengelbeck and F

    U. Schwengelbeck and F. H. M. Faisal, Phys. Rev. E 55 (1997) 6260

    work page 1997

  14. [14]

    Konkel and A

    S. Konkel and A. J. Makowski, Phys. Lett. A 238 (1998) 95

    work page 1998

  15. [15]

    O. F. de Alcantara Bonfim, J. Florencio, and F. C. S´ a Barreto, Phys. Rev. E 58 (1998) R2693

    work page 1998

  16. [16]

    O. F. de Alcantara Bonfim, J. Florencio, and F. C. S´ a Barreto, Phys. Rev. E 58 (1998) 6851

    work page 1998

  17. [17]

    O. F. de Alcantara Bonfim, J. Florencio, and F. C. S´ a Barreto, Phys. Lett. A (2000) 129

    work page 2000

  18. [18]

    J. A. de Sales and J. Florencio, Phys. Rev. E 67 (2003) 016216

    work page 2003

  19. [19]

    Bohm, Phys

    D. Bohm, Phys. Rev. 85 (1952) 166; 85 (1952) 180

    work page 1952

  20. [20]

    de Broglie, C

    L. de Broglie, C. R. Acad. Sci. Paris 183 (1926) 447; 185 (1927) 580

    work page 1926

  21. [21]

    de Broglie, Nonlinear Wave Mechanics (Elsevier, Am- sterdam, 1960)

    L. de Broglie, Nonlinear Wave Mechanics (Elsevier, Am- sterdam, 1960)

    work page 1960

  22. [22]

    M. W. Hersh and S. Smale, Differential Equations, Dy- namical Systems, and Linear Algebra (Academic Press, New York, 1974); P. G. Grazin, Nonlinear Systems (Cam- bridge University Press, Cambridge, 1993)

    work page 1974

  23. [23]

    Razavy, Am

    M. Razavy, Am. J. Phys. 48 (1980) 285

    work page 1980

  24. [24]

    N. G. van Kampen, J. Stat. Phys. 17 (1977) 71

    work page 1977