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 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.
Interference is commonly regarded as the most direct manifestation of the superposition principle. This association is natural for the linear Schrödinger equation, where coherent alternatives combine at the level of probability amplitudes. However, the situation becomes less transparent when nonlinear couplings are present, or when the field is only partially coherent. In this work, we argue that a more robust organizing principle is provided by the local flow generated by phase variations. In this sense, phase-induced flow acts as a unifying mechanism for interference-like dynamics in nonlinear and partially coherent Schrödinger systems. The discussion is developed from a hydrodynamic, or Bohmian, perspective, understood here as a practical probing tool rather than as an additional ontology. Three representative situations are considered: interfering Bose--Einstein condensates described by the Gross--Pitaevskii equation, nonlinear Schrödinger dynamics obtained by modifying the quantum-potential contribution, and partially coherent Airy beams described through their cross-spectral density. Although these systems differ in physical origin and mathematical implementation, they share a common dynamical structure: density-related observables are shaped by velocity fields determined by phase, or ensemble-phase, information. From this viewpoint, interference-like traits, localization, self-acceleration and coherence loss can be interpreted in terms of the preservation, deformation or breaking of the symmetries displayed by the underlying flow. This provides a compact way of connecting interference, nonlinear dynamics, classicality, coherence loss, and structured-light propagation within a single trajectory-based framework.
A classical limit of quantum dynamics can be defined by compensation of the quantum potential in the time-dependent Schrödinger equation. The quantum potential is a non-local quantity, defined in the trajectory-based form of the Schrödinger equation, due to Madelung, de Broglie, and Bohm, which formally generates the quantum-mechanical features in dynamics. Selective inclusion of the quantum potential for the degrees of freedom deemed "quantum," defines a hybrid quantum/classical dynamics, appropriate for molecular systems comprised of light and heavy nuclei. The wavefunction is associated with all of the nuclei, and the Ehrenfest, or mean-field, averaging of the force acting on the classical degrees of freedom, typical of the mixed quantum/classical methods, is avoided. The hybrid approach is used to examine evolution of light/heavy systems in the harmonic and double-well potentials, using conventional grid-based and approximate quantum-trajectory time propagation. The approximate quantum force is defined on spatial domains, which removes unphysical coupling of the wavefunction fragments corresponding to distinct classical channels or configurations. The quantum potential, associated with the quantum particle, generates forces acting on both quantum and classical particles to describe the backreaction.
We find that real and complex Bohmian quantum trajectories resulting from well-localized Klauder coherent states in the quasi-Poissonian regime possess qualitatively the same type of trajectories as those obtained from a purely classical analysis of the corresponding Hamilton-Jacobi equation. In the complex cases we treated the quantum potential results to a constant, such that the agreement is exact. For the real cases we provide conjectures for analytical solutions for the trajectories as well as the corresponding quantum potentials. The overall qualitative behavior is governed by the Mandel parameter determining the regime in which the wave functions evolve as solitonlike structures. We demonstrate these features explicitly for the harmonic oscillator and the P\"oschl-Teller potential.
A nonlinear quantum–classical transition wave equation is proposed for dissipative systems within the Caldirola-Kanai model. Equivalence of this transition equation to a scaled Schrödinger equation is proved. The dissipative dynamics is then studied in terms of what we call scaled trajectories following the standard procedure used in Bohmian mechanics. These trajectories depend on a continuous parameter allowing us a smooth transition from Bohmian to classical trajectories. Arrival times and actual momentum distribution functions are also analyzed. The propagation of a Gaussian wave packet in a viscid medium under the presence of constant, linear and harmonic potentials is studied. The gradual decoherence process and localization are easily visualized and understood within this theoretical framework.
This paper elucidates the dual structure of the Schrödinger dynamics in two correlated stages: (1) We first derive the real-valued Schrödinger equation from scratch without referring to classical mechanics, wave mechanics, nor optics, and thereby attain a concrete and clear interpretation of the Schrödinger (wave) function. Beginning with a factorization of the density distribution function of the particles to two component vectors in configuration space, we impose very simple conditions on them such as translational invariance of space-time and the conservation of flux under a given potential function. A real-valued path-integral is formulated as a Green function for the real-valued Schrödinger equation. (2) We then study a quantum stochastic path dynamics in a manner compatible with the Schrödinger equation. The relation between them is like the Langevin dynamics with the diffusion equation. Each quantum path describes a \textquotedblleft trajectory\textquotedblright\ in configuration space representing, for instance, a singly launched electron in the double-slit experiment that leaves a spot one by one at the measurement board, while accumulated spots give rise to the fringe pattern as predicted by the absolute square of the Schrödinger function. We start from the relationship between the Ito stochastic differential equation, the Feynman-Kac formula, and the associated parabolic partial differential equations, to one of which\ the Schrödinger equation is transformed. The physical significance of the quantum intrinsic stochasticity and the indirect correlation among the quantum paths and so on are discussed. The self-referential nonlinear interrelationship between the Schrödinger functions (regarded as a whole) and the quantum paths (as its parts) is identified as the ultimate mystery in quantum dynamics.
In this paper, the classical Schrödinger equation (CSE), which allows the study of classical dynamics in terms of wave functions, is analyzed theoretically and numerically. First, departing from classical (Newtonian) mechanics, and assuming an additional single-valued condition for the Hamilton’s principal function, the CSE is obtained. This additional assumption implies inherent non-classical features on the description of the dynamics obtained from the CSE: the trajectories do not cross in the configuration space. Second, departing from Bohmian mechanics and invoking the quantum-to-classical transition, the CSE is obtained in a natural way for the center of mass of a quantum system with a large number of identical particles. This quantum development imposes the condition of dealing with a narrow wave packet, which implicitly avoids the non-classical features mentioned above. We illustrate all the above points with numerical simulations of the classical and quantum Schrödinger equations for different systems.