Hydrodynamic Formulation Of Quantum Mechanics Research Articles

Instability and quantization in quantum hydrodynamics

We show how the quantum hydrodynamical formulation of quantum mechanics converts the nonlocality in the standard wave-like description of quantum systems by an instability of the quantum system, which opens the door to a new way for studying quantum systems based on known methodologies for studying the stability of fluids. As a second result, we show how the Madelung equations describe quantized energies without any external quantization conditions.

Geometric analogy between quantum dynamics and curved space through quantum hydrodynamics

In general relativity, the dynamics of objects is governed by the curvature of spacetime, which is caused by the presence of matter and energy. In contrast, in quantum mechanics, the dynamics is governed by the wavefunction, which completely describes the behavior of the particles. There is an ongoing effort to explore analogs of space and spacetime curvature in the context of quantum mechanics. Such analogies may reveal a deeper structure of quantum reality and its possible relations with Einstein’s theory of gravity. In this note, by coupling the non-relativistic Schrödinger equation with the heat equation and using the hydrodynamical formulation of quantum mechanics, we find that the dynamics of the particle is fully characterized by the normalized curvature of the wavefunction’s amplitude. Such a curvature obtains an analogy to the Ricci curvature of curved space in a Riemannian manifold. The proposed geometric correspondence provides a new pathway to explore quantum dynamics through the lens of differential geometry, the language of general relativity.

An appearance of classical matter from the self-organizing process of quantum systems

We present a quantum effect where matter follows the classical Hamilton-Jacobi equation, which emerges from quantum systems with Riemannian structures, as in standard quantum systems such as semiconductor heterostructures, quantum plasmas, and quantum dots. The proposed effect is derived from solving a standard elliptic partial differential equation of the radial part of the wave function, which is equivalent to a vanished quantum potential of the system. We then analyze such an effect and examine how the classical matter tends to be denser at the boundary region of the system when the quantum system is given in a finite region in space. While the proposed effect is derived from the hydrodynamical formulation of quantum mechanics, the results are free from any interpretation of quantum mechanics.

Uncertainty Relations in the Madelung Picture Including a Dissipative Environment.

In a recent paper, we have shown how in Madelung's hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment.

Dynamical Invariant for Dissipative Systems via Complex Quantum Hydrodynamics

For Hamiltonian systems with time-dependent potential, the Hamiltonian, and thus the energy, is no longer a constant of motion. However, for such systems as the parametric oscillator, i.e., an oscillator with time-dependent frequency ω(t), still, a dynamical invariant can be found that now has the dimension of action. The question, if such an invariant still exists after the addition of a dissipative friction force is analyzed for the classical as well as for the quantum mechanical case from different perspectives, particularly from that of a complex hydrodynamic formulation of quantum mechanics.

Diffusion Effect in Quantum Hydrodynamics

In this paper, we introduce (at least formally) a diffusion effect that is based on an axiom postulated by Werner Heisenberg in the early days of quantum mechanics. His statement was that—in quantum mechanics—kinematical quantities such as velocity must be treated as complex matrices. In the hydrodynamic formulation of quantum mechanics according to Madelung, the complex Schrödinger equation is rewritten in terms of two real equations—a continuity equation and a modified Hamilton–Jacobi equation. Considering seriously Heisenberg’s axiom, the velocity occurring in the continuity equation should be replaced by a complex one, thus introducing a diffusion term with an imaginary diffusion coefficient. Therefore, in quantum mechanics, there should be a diffusion effect showing up in the dynamics. This is the case in the time evolution of the free-motion wave packet under time reversal. The maximum returns to the initial position; however, the width of the wave packet does not shrink to its initial width. This effect is obvious but—as far as we know—it is not mentioned in any textbook. In our paper, we discuss this effect in detail and show the connection with a complex version of quantum hydrodynamics.

Quantum hydrodynamics with complex quantities

The hydrodynamic formulation of quantum mechanics is extended to include further densities known in hydrodynamics and their time-evolution.The equations of motion of the complex Bohmian quantities in position space are used to obtain transport equations for complex momentum and energy densities that can be unified with the continuity equation. Separation into real and imaginary parts leads to hydrodynamic equations known in the literature. The same procedure is repeated in momentum space. Finally, the transition from transport equations describing local densities to Euler equations in a frame co-moving with the convection velocity is made.

Vortex structures in photodetachment by few-cycle circularly polarized pulses

Generation of electron vortices in photodetachment of ${\mathrm{H}}^{\ensuremath{-}}$ by circularly polarized laser pulses is analyzed by means of strong-field approximation and by numerically solving the time-dependent Schr\"odinger equation. A very good agreement is shown for the magnitude and the phase of the probability amplitude of photodetachment from both approaches. We demonstrate that spiral-like patterns in the probability amplitude of detachment, observed for a pair of counter-rotating circularly polarized laser pulses, cannot be associated with nonvanishing topological charge vortices. The latter can be generated, on the other hand, by a circularly polarized laser pulse or a sequence of such pulses with corotating polarizations. Such interpretation of our results follows from the hydrodynamical formulation of quantum mechanics and its generalization to arbitrary parametric spaces.

Trajectory-based modelling of the quantum dynamics of vibrational predissociation: Application to the Ar [formula omitted] complex

The quantum dynamics of vibrational predissociation of the Ar ⋯Br2 triatomic molecule is described within a trajectory-based framework. The Br2 stretching mode is mapped into a set of classical (coupled) harmonic oscillators, associated to each vibrational state of the diatomic molecule. The time evolution of the molecular wave packet along the dissociation coordinate is described within the hydrodynamical formulation of quantum mechanics, specifically using the interacting trajectory representation. The relatively small number of interacting trajectories required to attain numerical convergence (N=100), makes the present model very appealing in comparison with other trajectory-based methods. The underlying parameterisation of the density was found to represent accurately the evolution of the projection of the molecular wave packet along the van der Waals mode, from the ground vibrational state into the continuum. The computed lifetime of the predissociating level and the population dynamics are in very good agreement with those observed experimentally.

Quantum Interference, Hidden Symmetries: Theory and Experimental Facts

The concept of quantum superposition is reconsidered and discussed from the viewpoint of Bohmian mechanics, the hydrodynamic formulation of quantum mechanics, in order to elucidate some physical consequences that go beyond the simple mathematical idea of linearly combining vectors in a Hilbert space. Specifically, the discussion turns around the connection between symmetries characterizing the wave function and the behavior in time displayed by the quantum flux when the latter is analyzed in terms of streamlines (Bohmian trajectories). This is illustrated with a series of analytical results and numerical simulations, which include Young’s two-slit experiment, counter-propagating wave packets, grating diffiraction and quantum carpets (e.g., Talbot carpets), and diffiraction under confinement conditions. From the analysis presented it follows that quantum paradoxes appear whenever symmetries related to interference are neglected in the interpretation and understanding of the corresponding phenomena.

Reconstruction of Bohmian trajectories via weak measurement for bipartite states

Here we propose a protocol to reconstruct Bohmian trajectories for bipartite states of massive particles moving in an arbitrary one-dimensional potential. The proposal employs weak measurements and generalizes recent proposals presented in the literature (Kocsis et al., 2011). Finally, our work opens perspectives to experimentally study quantum correlation (primary resource for quantum information tasks) using the hydrodynamic formulation of quantum mechanics as suggested by Madelung and Bohm.

Description of Classical and Quantum Interference in View of the Concept of Flow Line

Bohmian mechanics, a hydrodynamic formulation of quantum mechanics, relies on the concept of trajectory, which evolves in time in compliance with dynamical information conveyed by the wave function. Here this appealing idea is considered to analyze both classical and quantum interference, thus providing an alternative and more intuitive framework to understand the time-evolution of waves, either in terms of the flow of energy (for mechanical waves, sound waves, electromagnetic waves, for instance) or, analogously, the flow of probability (quantum waves), respectively. Furthermore, this procedure also supplies a more robust explanation of interference phenomena, which currently is only based on the superposition principle. That is, while this principle only describes how different waves combine and what effects these combinations may lead to, flow lines provide a more precise explanation on how the energy or probability propagate in space before, during and after the combination of such waves, without dealing with them separately (i.e., the combination or superposition is taken as a whole). In this sense, concepts such as constructive and destructive interference, typically associated with the superposition principle, physically correspond to more or less dense swarms of (energy or probability) flow lines, respectively. A direct consequence of this description is that, when considering the distribution of electromagnetic energy flow lines behind two slits, each one covered by a differently oriented polarizer, it is naturally found that external observers' information on the slit crossed by single photons (understood as energy parcels) is totally irrelevant for the existence of interference fringes, in striking contrast with what is commonly stated and taught.

Gaussian distributions, Jacobi group, and Siegel-Jacobi space

Let N be the space of Gaussian distribution functions over ℝ, regarded as a 2-dimensional statistical manifold parameterized by the mean μ and the deviation σ. In this paper, we show that the tangent bundle of N, endowed with its natural Kähler structure, is the Siegel-Jacobi space appearing in the context of Number Theory and Jacobi forms. Geometrical aspects of the Siegel-Jacobi space are discussed in detail (completeness, curvature, group of holomorphic isometries, space of Kähler functions, and relationship to the Jacobi group), and are related to the quantum formalism in its geometrical form, i.e., based on the Kähler structure of the complex projective space. This paper is a continuation of our previous work [M. Molitor, “Remarks on the statistical origin of the geometrical formulation of quantum mechanics,” Int. J. Geom. Methods Mod. Phys. 9(3), 1220001, 9 (2012); M. Molitor, “Information geometry and the hydrodynamical formulation of quantum mechanics,” e-print arXiv (2012); M. Molitor, “Exponential families, Kähler geometry and quantum mechanics,” J. Geom. Phys. 70, 54–80 (2013)], where we studied the quantum formalism from a geometric and information-theoretical point of view.

Reconstruction of Bohm trajectories and wave functions from interferometric measurements

We propose a generalization of the technique of quantum state reconstruction to obtain the space derivative of the phase of a wave function, which is central to the hydrodynamic formulation of quantum mechanics in the manner of Madelung and Bohm. In this way we can construct Bohm trajectories and make contact with a recent experiment [Kocsis et al., Science 332, 1170 (2011)] extracting these trajectories after a double slit.

Equations of motion for a time-dependent open system: An algebraic approach

An algebraic approach based on the concept of local densities is introduced in order to provide an alternative derivation of several equations central to the hydrodynamical formulation of quantum mechanics. The origin of this work lays in an algebraic equation which is built based on the concept of quantum densities. This enables us to derive the regional and local forms of several significant quantum laws and equations, namely Heisenberg equation of motion, hypervirial theory and quantum Navier–Stokes equation. In particular, atomic force law and local virial theorem for a time-dependent open system are extracted without referring to rigorous Schwinger’s principle of stationary action.

Hydrodynamical description of strong field-driven electron dynamics

We present the implementation of the Bohmian hydrodynamical formulation of quantum mechanics to the description of strong field-driven electron dynamics. We consider the case of one-dimensional hydrogenic atoms subjected to strong and short laser pulses as well as one-electron ion-atom collisions. The quantum trajectories clearly picture the dynamics and yield an accurate description of the observables.

Nonlinear quantum mechanics, complex classical mechanics and conservation laws for closed and open systems

Phase and amplitude of the complex wave function are not independent of each other, but coupled, which becomes obvious looking at Madelung's hydrodynamic formulation of quantum mechanics. In the time-independent case, this leads to a kind of conservation law that allows for the reformulation of the linear Schrödinger equation in terms of a nonlinear Ermakov equation which is equivalent to a complex Riccati equation where the quadratic term in this equation explains the origin of the phase-amplitude coupling. A similar conservation law and corresponding nonlinear equations can also be found in the time-dependent case. The gain from the nonlinear formulations emerges when open systems with dissipation and irreversibility are considered. Describing this kind of systems by an effective nonlinear Schrödinger equation leads to a modification of the above-mentioned equations with new qualitative effects like Hopf bifurcations.

Bohmian picture of laser-atom interaction

We present the implementation of the Bohmian hydrodynamic formulation of quantum mechanics to the description of electron dynamics in one-dimensional laser-atom interaction. The resulting quantum trajectories clearly picture the dynamics and yield an accurate description of observables.

Hydrodynamic perspective on memory in time-dependent density-functional theory

The adiabatic approximation of time-dependent density-functional theory is studied in the context of nonlinear excitations of two-electron singlet systems. We compare the exact time evolution of these systems to the adiabatically exact one obtained from time-dependent Kohn-Sham calculations relying on the exact ground-state exchange-correlation potential. Thus, we can show under which conditions the adiabatic approximation breaks down and memory effects become important. The hydrodynamic formulation of quantum mechanics allows us to interpret these results and relate them to dissipative effects in the Kohn-Sham system. We show how the breakdown of the adiabatic approximation can be inferred from the rate of change of the ground-state noninteracting kinetic energy.

PARALLEL ADAPTIVE QUANTUM TRAJECTORY METHOD FOR WAVEPACKET SIMULATIONS

Time-dependent wavepackets are widely used to model various phenomena in physics. One approach in simulating the wavepacket dynamics is the quantum trajectory method (QTM). Based on the hydrodynamic formulation of quantum mechanics, the QTM represents the wavepacket by an unstructured set of pseudoparticles whose trajectories are coupled by the quantum potential. The governing equations for the pseudoparticle trajectories are solved using a computationally-intensive moving weighted least squares (MWLS) algorithm, and the trajectories can be computed in parallel. This paper contributes a strategy for improving the performance of wavepacket simulations using the QTM. Specifically, adaptivity is incorporated into the MWLS algorithm, and loop scheduling techniques are employed to dynamically load balance the parallel computation of the trajectories. The adaptive MWLS algorithm reduces the amount of computations without sacrificing accuracy, while adaptive loop scheduling addresses the load imbalance introduced by the algorithm and the runtime system. Results of experiments on a Linux cluster are presented to confirm that the adaptive MWLS reduces the trajectory computation time by up to 24%, and adaptive loop scheduling achieves parallel efficiencies of up to 85% when simulating a free particle.