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

arXiv:2007.02176 · Physical Review A

Quantum particles that behave as free classical particles

Sergio A. Hojman, Felipe A. Asenjo

Published Nov 9, 2020• 9 citations
arXivHTMLPDF

Abstract

The existence of nonvanishing Bohm potentials, in the Madelung-Bohm version of the Schr\"odinger equation, allows for the construction of particular solutions for states of quantum particles interacting with nontrivial external potentials that propagate as free classical particles. Such solutions are constructed with phases which satisfy the classical Hamilton-Jacobi for free particles and whose probability densities propagate with constant velocity, as free classical particles do.

DOI
10.1103/physreva.102.052211

Related papers

A new approach to solving the Schrödinger equation

Sergio A. Hojman, Felipe A. Asenjo

A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non--vanishing Bohm potentials. New examples of this kind are found and discussed.

arXiv:2007.02182•Jul 4, 2020•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

Exact computation of quantum wave functions for nonlinear potentials

Winfried Loghmiller, Jean-Jacques Slotine

Recent work shows that the Schrödinger equation can be solved exactly based only on classical least action. The computation is based on first computing the multi-valued action solutions of the Hamilton-Jacobi equation, computing the classical propagated density along all stationary action multipaths, and finally constructing the exact kernel and wave function based on these classical multi-valued quantities alone. The method requires that the propagated classical square root density along each stationary action path is harmonic, as is the case for basic examples such as e.g., the double-slit, quantum tunneling, or the Einstein-Podolsky-Rosen experiment. For more complex cases, residual Bohm-like quantum potentials along each stationary action path can be eliminated through a time-rescaling. This paper proposes an extended version of the above construction which avoids the need for time-rescaling even in the most general cases. This is achieved by introducing in the Hamilton-Jacobi equation a real viscosity term, which classically simply corresponds to replacing a lumped parameter model with a continuum model including a volumetric stress tensor. This keeps the earlier construction unchanged, except that the real stationary action multipaths are now real stationary viscous action multipaths defined with respect to an associated modified Lagrangian. As in the earlier version, the construction extends straightforwardly to the special and general relativistic contexts, thus describing in a common language the quantum, classical, and relativistic cases. Note that the global Bohm potential term in the lumped Hamilton-Jacobi equation associated with a Madelung transform does not have a classical equivalent, while the volumetric stress tensor term in our viscous Hamilton-Jacobi equation directly represents the real paths of a classical deformed particle field.

quant-pharXiv:2609.14480•Sep 15, 2026•0 citations

Relating a particle in a constant gravitational field and the harmonic oscillator in the classical and quantum mechanics frameworks

Gilberto Silva-Ortigoza, Emmanuel Paz-Córdova, Israel Julián-Macías +5 more

Abstract In the present work we use a point transformation defined in the extended configuration space to show that, in both classical and quantum formalisms, the evolution of a particle under the influence of a constant gravitational field can be related to the evolution of the harmonic oscillator. That is, in the classical mechanics regime, we show that this transformation allows us to establish the connection in the Newtonian, Lagrangian, Hamilton-Jacobi and Hamiltonian formalisms. In the quantum mechanics regime, the transformation determines a relationship between the wave functions, solutions to the corresponding Schrödinger equations, the propagators and the Wigner functions. Furthermore, by using the quantum potential approach to quantum mechanics, we show that the classical point transformation, in a natural manner, induces a quantum point transformation in the corresponding extended quantum configuration space, that determines the connection between the corresponding Madelung-Bohm trajectories. We find that the quantum transformation has the same functional form as the classical one even though the quantum or Madelung-Bohm trajectories are in general totally different from the classical ones. Then, we show that the general results can also be applied to the corresponding paraxial wave equations, and by using the stationary solutions of the quantum Harmonic oscillator and the Airy solution obtained by Berry and Balazs, as two applications, we determine the evolution of the Hermite-Gaussian and Airy beams in free space, in a linear and in a quadratic optical media. Finally, by using the point transformation, we construct two metrics whose geodesics correspond precisely to the classical trajectories for the harmonic oscillator and a particle in a uniform gravitational field.

Physica Scripta•Dec 1, 2025•0 citations

Schrodinger Equation and Free Particle Wave Function

Karl De Paepe

Using the wave function of a free particle we obtain a solution of the Schrodinger equation for a class of potentials.

•Jan 1, 2021•0 citations

Quantum and Classical Mechanics of Hamiltonian Systems Having Exponential-Type Potentials

J. Akande, L. H. Koudahoun, Kolawolé Kêgnidé Damien Adjaï +2 more

The problem to solve consists of finding explicit solutions for the classical equation of motion of a particle subject to an exponential-type potential and for the quantum version.

•Sep 1, 2016•0 citations
The existence of nonvanishing Bohm potentials, in the Madelung-Bohm version of the Schr"odinger equation, allows for the construction of particular solutions for states of quantum particles interacting with nontrivial external potentials that propagate as free classical particles. Such solutions are constructed with phases which satisfy the classical Hamilton-Jacobi for free particles and whose probability densities propagate with constant velocity, as free classical particles do.
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non--vanishing Bohm potentials. New examples of this kind are found and discussed.
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.
Recent work shows that the Schrödinger equation can be solved exactly based only on classical least action. The computation is based on first computing the multi-valued action solutions of the Hamilton-Jacobi equation, computing the classical propagated density along all stationary action multipaths, and finally constructing the exact kernel and wave function based on these classical multi-valued quantities alone. The method requires that the propagated classical square root density along each stationary action path is harmonic, as is the case for basic examples such as e.g., the double-slit, quantum tunneling, or the Einstein-Podolsky-Rosen experiment. For more complex cases, residual Bohm-like quantum potentials along each stationary action path can be eliminated through a time-rescaling. This paper proposes an extended version of the above construction which avoids the need for time-rescaling even in the most general cases. This is achieved by introducing in the Hamilton-Jacobi equation a real viscosity term, which classically simply corresponds to replacing a lumped parameter model with a continuum model including a volumetric stress tensor. This keeps the earlier construction unchanged, except that the real stationary action multipaths are now real stationary viscous action multipaths defined with respect to an associated modified Lagrangian. As in the earlier version, the construction extends straightforwardly to the special and general relativistic contexts, thus describing in a common language the quantum, classical, and relativistic cases. Note that the global Bohm potential term in the lumped Hamilton-Jacobi equation associated with a Madelung transform does not have a classical equivalent, while the volumetric stress tensor term in our viscous Hamilton-Jacobi equation directly represents the real paths of a classical deformed particle field.
Abstract In the present work we use a point transformation defined in the extended configuration space to show that, in both classical and quantum formalisms, the evolution of a particle under the influence of a constant gravitational field can be related to the evolution of the harmonic oscillator. That is, in the classical mechanics regime, we show that this transformation allows us to establish the connection in the Newtonian, Lagrangian, Hamilton-Jacobi and Hamiltonian formalisms. In the quantum mechanics regime, the transformation determines a relationship between the wave functions, solutions to the corresponding Schrödinger equations, the propagators and the Wigner functions. Furthermore, by using the quantum potential approach to quantum mechanics, we show that the classical point transformation, in a natural manner, induces a quantum point transformation in the corresponding extended quantum configuration space, that determines the connection between the corresponding Madelung-Bohm trajectories. We find that the quantum transformation has the same functional form as the classical one even though the quantum or Madelung-Bohm trajectories are in general totally different from the classical ones. Then, we show that the general results can also be applied to the corresponding paraxial wave equations, and by using the stationary solutions of the quantum Harmonic oscillator and the Airy solution obtained by Berry and Balazs, as two applications, we determine the evolution of the Hermite-Gaussian and Airy beams in free space, in a linear and in a quadratic optical media. Finally, by using the point transformation, we construct two metrics whose geodesics correspond precisely to the classical trajectories for the harmonic oscillator and a particle in a uniform gravitational field.