The hydrodynamic representation of quantum mechanics describes virtual flow as if a quantum system were fluid in motion. This formulation illustrates pointlike vortices when the phase of a wavefunction becomes nonintegrable at nodal points. We study the dynamics of such pointlike vortices in the hydrodynamic representation for a two-particle wavefunction. In particular, we discuss how quantum entanglement influences vortex-vortex dynamics. For this purpose, we employ the time-dependent quantum variational principle combined with the Rayleigh-Ritz method. We analyze the vortex dynamics and establish connections with Dirac's generalized Hamiltonian formalism.
Exact analytic solutions of the time dependent Schrodinger equation are produced that exhibit a variety of vortex structures. The qualitative analysis of the motion of vortex lines is presented and various types of vortex behavior are identified. Vortex creation and annihilation and vortex interactions are illustrated in the special cases of the free motion, the motion in the harmonic potential, and in the constant magnetic field. Similar analysis of the vortex motions is carried out also for a relativistic wave equation.
Engineering of synthetic magnetic flux in Bose-Einstein condensates [Lin et al., Nature {\bf 462}, 628 (2009)] has prospects for attaining the high vortex densities necessary to emulate the fractional quantum Hall effect. We analytically establish the hydrodynamical behaviour of a condensate in a uniform synthetic magnetic field, including its density and velocity profile. Importantly, we find that the onset of vortex nucleation observed experimentally corresponds to a dynamical instability in the hydrodynamical solutions and reveal other routes to instability and anticipated vortex nucleation…
Physical vacuum is a special superfluid medium populated by enormous amount of virtual particle-antiparticle pairs. Its motion is described by the modified Navier-Stokes equation: (a)~the pressure gradient divided by the mass density is replaced by the gradient from the quantum potential; (b)~time-averaged the viscosity vanishes, but its variance is not zero. Vortex structures arising in this medium show infinitely long lifetime owing to zero average viscosity. The nonzero variance is conditioned by exchanging the vortex energy with zero-point vacuum fluctuations. The vortex has a non-zero cor…
We extend our previous analysis of the motion of vortex lines [I. Bialynicki-Birula, Z. Bialynicka-Birula and C. Sliwa, Phys. Rev. A 61, 032110 (2000)] from linear to a nonlinear Schroedinger equation with harmonic forces. We also argue that under certain conditions the influence of the contact nonlinearity on the motion of vortex lines is negligible. The present analysis adds new weight to our previous conjecture that the topological features of vortex dynamics are to a large extent universal.
We show that zero-energy flows appear in many particle systems as same as in single particle cases in 2-dimensions. Vortex patterns constructed from the zero-energy flows can be investigated in terms of the eigenstates in conjugate spaces of Gel'fand triplets. Stable patterns are written by the superposition of zero-energy eigenstates. On the other hand vortex creations and annihilations are described by the insertions of unstable eigenstates with complex-energy eigenvalues into the stable patterns. Some concrete examples are presented in the 2-dimensional parabolic potential barrier case. %, …
Using the linear local induction approximation, we investigated the self-induced motion of a vortex line that corresponds to the motion of a particle in quantum mechanics. Concerning Kelvin waves, the effective Schrödinger equation, physical quantities operators, and the corresponding path-integral formula are obtained. The vortex line-particle mapping may help in understanding particle motion in quantum mechanics.
The Schrodinger equation can be derived using the minimum Fisher information principle. I discuss why such an approach should work, and also show that the Kahler and Hilbert space structures of quantum mechanics result from combining the symplectic structure of the hydrodynamical model with the Fisher information metric.
Generalization of the Kutta-Joukowski theorem for the hydrodynamic forces acting on a quantized vortex
Université Libre de Bruxelles（布鲁塞尔自由大学）；Observatoire de Paris, PSL Research University, CNRS, Université Paris Diderot, Sorbonne Paris Cité（巴黎天文台，PSL研究大学，法国国家科学研究中心，巴黎第七大学，索邦巴黎西岱大学）
It is shown that the Bohmian mechanics and the Madelung quantum hydrodynamics are different theories and the latter is a better ontological interpretation of quantum mechanics. A new stochastic interpretation of quantum mechanics is proposed, which is the background of the Madelung quantum hydrodynamics. Its relation to the complex mechanics is also explored. A new complex hydrodynamics is proposed, which eliminates completely the Bohm quantum potential. It describes the quantum evolution of the probability density by a convective diffusion with imaginary transport coefficients.
Electron vortex beams offer unique opportunities for the study of chiral or magnetic structures in electron microscopes and of fundamental effects of quantum interference in particle physics. Immersing a cathode in a solenoid field presents a highly efficient and flexible method for the generation of vortex electron beams which is utilized at accelerators, but has not yet been realized in an electron microscope. The conditions for the generation of vortex beams with quantized orbital angular momentum from an immersed cathode in an electron microscope are discussed, and general possibilities of…
Quantum turbulence that exhibits vortex creation, annihilation and interactions is demonstrated as an exact solution of the time-dependent, free-particle Schrödinger equation evolved from a smooth random-phased initial condition. Relaxed quantum turbulence in 2D and 3D exhibits universal scaling in the steady-state energy spectrum as k-1 in small scales. Due to the lack of dissipation, no evidence of the Kolmogorov-type forward energy cascade in 3D or the inverse energy cascade in 2D is found, but the rotational and potential flow components do approach equi-partition in the scaling regime. In…
A stochastic Euler equation is proposed, describing the motion of a particle density, forced by the random action of virtual photons in vacuum. After time averaging, the Euler equation is reduced to the Reynolds equation, well studied in turbulent hydrodynamics. It is shown that the average pressure is nonlocal and the magnitude of the turbulent flow obeys the Fick law. Using the Madelung transformation, the Schrodinger equation is derived without any other assumptions.
Motivated by recent progress in the generation of optical spatiotemporal vortex pulses (STVPs), there is a theoretical discussion about the transverse orbital angular momentum (OAM) carried by such pulses. Two recent works [K. Y. Bliokh, Phys. Rev. Lett. 126, 243601 (2021)] and [S. W. Hancock et al., Phys. Rev. Lett. 127, 193901 (2021)] claimed the OAM values which differ by a factor of 2 for circular STVPs. Here we resolve this controversy by showing that the result by Hancock et al. is correct for the total OAM, while the result by Bliokh describes the suitably defined intrinsic part of the …
Quantum hydrodynamics is a formulation of quantum mechanics based on the probability density and flux (current) density of a quantum system. It can be used to define trajectories which allow for a particle-based interpretation of quantum mechanics, commonly known as Bohmian mechanics. However, quantum hydrodynamics rests on the usual time-dependent formulation of quantum mechanics where time appears as a parameter. This parameter describes the correlation of the state of the quantum system with an external system -- a clock -- which behaves according to classical mechanics. With the Exact Fact…
A quantum mechanical description of vortices in two dimensional superfluid $^4$He flims is presented. Single vortex creation and annihilation operators are defined and wavefunctions for these states are explicitly constructed. A hamitonian for these states is then proposed.
The operator equations for quantum hydrodynamics are discussed and solved in a simple cylindrical geometry. We find a solution with the velocity curl "frozen" into a density of the liquid in the absence of singular vortex lines. The spectrum of small oscillations around this solution is found as a generalization of the standard phonon spectrum, and the stability of the solution is demonstrated.
We consider a Finslerian type geometrization of the non-relativistic quantum mechanics in its hydrodynamical (Madelung) formulation, by also taking into account the effects of the presence of the electromagnetic fields on the particle motion. In the Madelung representation the Schrödinger equation can be reformulated as the classical continuity and Euler equations of classical fluid mechanics in the presence of a quantum potential, representing the quantum hydrodynamical evolution equations. The equation of particle motion can then be obtained from a Lagrangian similar to its classical counter…
