Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physics Letters A

A Bohmian trajectory analysis of singular wave functions

Ángel S. Sanz, Luis Lorenzo Sánchez-Soto, Andrea Aiello

Published Mar 7, 2024• 2 citations
Publisher

Abstract

The Schrödinger equation admits smooth and finite solutions that spontaneously evolve into a singularity, even for a free particle. This blowup is generally ascribed to the intrinsic dispersive character of the associated time evolution. We resort to the notion of quantum Bohmian trajectories to relate this singular behavior to local phase variations, which generate an underlying velocity field responsible for driving the quantum flux toward the singular region.

DOI
10.1016/j.physleta.2024.129428

Related papers

Symmetries and singular behaviors with Bohmian trajectories

Ángel S. Sanz

Abstract Quantum mechanics is able to predict challenging behaviors even in the simplest physical scenarios. These behaviors are possible because of the important dynamical role that phase plays in the evolution of quantum systems, and are very similar, on the other hand, to effects observable in analogous optical systems. This work focuses on how Bohmian mechanics proves to be a rather convenient theoretical framework to analyze phase-based phenomena, since the phase constitutes the central element in this hydrodynamical formulation of quantum mechanics. More specifically, it allows us to understand how spatial phase variations give rise to velocity fields that eventually rule the dynamical behavior of quantum systems, and that, when integrated in time locally (i.e., taking into account specific positions), they provide us with a neat local (point by point) description of the system evolution in the configuration space. Indeed, it will also be seen that this idea transcends the quantum realm and can be profitably used to describe the behavior of optical analogs with rather singular behaviors. With this purpose, two interesting phenomena that take place in free space are considered, namely, the self-acceleration and shape-invariance of Airy beams, and spontaneous self-focusing.

arXiv:2407.08096•Journal of Physics Conference Series•Nov 1, 2024•3 citations

Global existence and uniqueness of Bohmian trajectories

Karin Berndl

It is shown that Bohmian mechanics is internally consistent in the sense that the equations of motion typically have global solutions despite the fact that the velocity field is singular at the nodes of the wave function and at other points. This result is fundamental for the derivation of the quantum formalism. The role of the quantum flux is emphasized.

arXiv:quant-ph/9509009•Sep 18, 1995•0 citations

Bohmian trajectories of the time-oscillating Schrödinger equations

Dandan Li, Jinqiao Duan, Lin Li +1 more

Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper, we study the Schrödinger equation with a rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in a measure, and the limit coincides with the Bohmian trajectory for the effective Schrödinger equation on a finite time interval. This is beneficial for efficient simulation of the Bohmian trajectories in oscillating potential fields.

arXiv:2109.04039•Chaos An Interdisciplinary Journal of Nonlinear Science•Oct 1, 2021•1 citation

Non-differentiable Bohmian trajectories

Gebhard Grübl, Markus Penz

A solution $ψ$ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field $\hbar \Im \left( \bigtriangledown ψ/mψ\right) .$ In the case of one specific non-differentiable weak solution $Ψ$ we show how Bohmian trajectories can be obtained for $Ψ$ from the trajectories of a sequence $Ψ_{n}\rightarrow Ψ.$ (For any real $t$ the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors $Ψ$ in the Hilbert space on which the Schrödinger evolution $% Ψ\mapsto e^{-iht}Ψ$ acts.

arXiv:1011.2852•Nov 12, 2010•1 citation

Exploring the nonclassical dynamics of the “classical” Schrödinger equation

David Navia, Ángel S. Sanz

The introduction of nonlinearities in the Schrödinger equation has been considered in the literature as an effective manner to describe the action of external environments or mean fields. Here, in particular, we explore the nonlinear effects induced by subtracting a term proportional to Bohm’s quantum potential to the usual (linear) Schrödinger equation, which generates the so-called “classical” Schrödinger equation. Although a simple nonlinear transformation allows us to recover the well-known classical Hamilton–Jacobi equation, by combining a series of analytical results (in the limiting cases) and simulations (whenever the analytical treatment is unaffordable), we find an analytical explanation to why the dynamics in the nonlinear “classical” regime is still strongly nonclassical. This is even more evident by establishing a one-to-one comparison between the Bohmian trajectories associated with the corresponding wave function and the classical trajectories that one should obtain. Based on these observations, it is clear that the transition to a fully classical regime requires extra conditions in order to remove any trace of coherence, which is the truly distinctive trait of quantum mechanics. This behavior is investigated in three paradigmatic cases, namely, the dispersion of a free propagating localized particle, the harmonic oscillator, and a simplified version of Young’s two-slit experiment.

arXiv:2312.02977•Annals of Physics•Mar 4, 2024•5 citations

An essay on semiclassical analysis for microlocal singularities, turbulence intensity and integration of singularities by Schrödinger equation in probabilistic behavior

Jorge Henrique de Oliveira Sales, Rômulo Damasclin Chaves dos Santos

The Schrödinger equation governs the probabilistic behavior of quantum particles through the wave function. Microlocal singularities denote regions with significantly high probability density or abrupt changes therein. By visualizing the probability distribution in time and space, we discern regions with higher probability density, indicative of potential microlocal singularities. These regions probably correspond to areas with a greater probability of particle presence. Such analysis aligns with Theorem 1, predicting characteristics of microlocal singularities of wave functions. Furthermore, Theorem 2 postulates that semiclassical path integrals along these singularities contribute significantly to solving the Schrödinger equation. Interpreting the temporal evolution of the probability density in the probability distribution visualization reveals the propagation of the particle over time. Regions of high density mean likely presence of particles at specific times, aligning with the predictions of Theorem 2. Consequently, the analysis of the contribution of high-density regions to the temporal evolution of the wave function resembles semi-classical path integral calculations. Thus, our findings demonstrate that visualization of probability distributions obtained from the numerical resolution of the Schrödinger equation allows a comprehensive interpretation of the behavior of quantum particles, consistent with the theorems.

OBSERVATÓRIO DE LA ECONOMÍA LATINOAMERICANA•May 20, 2024•0 citations
The Schrödinger equation admits smooth and finite solutions that spontaneously evolve into a singularity, even for a free particle. This blowup is generally ascribed to the intrinsic dispersive character of the associated time evolution. We resort to the notion of quantum Bohmian trajectories to relate this singular behavior to local phase variations, which generate an underlying velocity field responsible for driving the quantum flux toward the singular region.
Abstract Quantum mechanics is able to predict challenging behaviors even in the simplest physical scenarios. These behaviors are possible because of the important dynamical role that phase plays in the evolution of quantum systems, and are very similar, on the other hand, to effects observable in analogous optical systems. This work focuses on how Bohmian mechanics proves to be a rather convenient theoretical framework to analyze phase-based phenomena, since the phase constitutes the central element in this hydrodynamical formulation of quantum mechanics. More specifically, it allows us to understand how spatial phase variations give rise to velocity fields that eventually rule the dynamical behavior of quantum systems, and that, when integrated in time locally (i.e., taking into account specific positions), they provide us with a neat local (point by point) description of the system evolution in the configuration space. Indeed, it will also be seen that this idea transcends the quantum realm and can be profitably used to describe the behavior of optical analogs with rather singular behaviors. With this purpose, two interesting phenomena that take place in free space are considered, namely, the self-acceleration and shape-invariance of Airy beams, and spontaneous self-focusing.
Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper, we study the Schrödinger equation with a rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in a measure, and the limit coincides with the Bohmian trajectory for the effective Schrödinger equation on a finite time interval. This is beneficial for efficient simulation of the Bohmian trajectories in oscillating potential fields.
A solution ψψψ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field ℏℑ(▽ψ/mψ).\hbar \Im \left( \bigtriangledown ψ/mψ\right) .ℏℑ(▽ψ/mψ). In the case of one specific non-differentiable weak solution ΨΨΨ we show how Bohmian trajectories can be obtained for ΨΨΨ from the trajectories of a sequence Ψn→Ψ.Ψ_{n}\rightarrow Ψ.Ψn​→Ψ. (For any real ttt the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors ΨΨΨ in the Hilbert space on which the Schrödinger evolution % Ψ\mapsto e^{-iht}Ψ acts.
The introduction of nonlinearities in the Schrödinger equation has been considered in the literature as an effective manner to describe the action of external environments or mean fields. Here, in particular, we explore the nonlinear effects induced by subtracting a term proportional to Bohm’s quantum potential to the usual (linear) Schrödinger equation, which generates the so-called “classical” Schrödinger equation. Although a simple nonlinear transformation allows us to recover the well-known classical Hamilton–Jacobi equation, by combining a series of analytical results (in the limiting cases) and simulations (whenever the analytical treatment is unaffordable), we find an analytical explanation to why the dynamics in the nonlinear “classical” regime is still strongly nonclassical. This is even more evident by establishing a one-to-one comparison between the Bohmian trajectories associated with the corresponding wave function and the classical trajectories that one should obtain. Based on these observations, it is clear that the transition to a fully classical regime requires extra conditions in order to remove any trace of coherence, which is the truly distinctive trait of quantum mechanics. This behavior is investigated in three paradigmatic cases, namely, the dispersion of a free propagating localized particle, the harmonic oscillator, and a simplified version of Young’s two-slit experiment.
The Schrödinger equation governs the probabilistic behavior of quantum particles through the wave function. Microlocal singularities denote regions with significantly high probability density or abrupt changes therein. By visualizing the probability distribution in time and space, we discern regions with higher probability density, indicative of potential microlocal singularities. These regions probably correspond to areas with a greater probability of particle presence. Such analysis aligns with Theorem 1, predicting characteristics of microlocal singularities of wave functions. Furthermore, Theorem 2 postulates that semiclassical path integrals along these singularities contribute significantly to solving the Schrödinger equation. Interpreting the temporal evolution of the probability density in the probability distribution visualization reveals the propagation of the particle over time. Regions of high density mean likely presence of particles at specific times, aligning with the predictions of Theorem 2. Consequently, the analysis of the contribution of high-density regions to the temporal evolution of the wave function resembles semi-classical path integral calculations. Thus, our findings demonstrate that visualization of probability distributions obtained from the numerical resolution of the Schrödinger equation allows a comprehensive interpretation of the behavior of quantum particles, consistent with the theorems.