Droplets moving on a fluid surface: interference pattern from two slits

The Feynman path integral approach for solving the motion of a droplet along a silicon oil surface is developed by replacing the Planck constant by a surrogate parameter. The latter is proportional to the surface tension of the silicon oil multiplied by the area of the thin air film, separating the droplet from the oil, and by the half-period of the Faraday oscillations. It is shown that the Navier-Stokes equation together with the mass conservation equation can be reduced to the Schr\"{o}dinger equation when the surrogate parameter replaces the Planck constant. The Feynman path integral underlying the Schr\"{o}dinger equation is used then to calculate a wave function that plays the role of the de Broglie pilot-wave.Comment: 8 pages, 6 figure

N-Slit Interference: Fractals in Near-Field Region, Bohmian Trajectories

Scattering cold particles on an $N$-slit grating is shown to reproduce an interference pattern, that manifests itself in the near-field region as the fractal Talbot carpet. In the far-field region the pattern is transformed to an ordinary diffraction, where principal beams are partitioned from each other by ($N-2$) weak ones. A probability density plot of the wave function, to be represented by a gaussian wavepacket, is calculated both in the near-field region and in the far-field one. Bohmian (geodesic) trajectories, to be calculated by a guidance equation, are superimposed on the probability density plot well enough. It means, that a particle, moving from a source to a detector, passes across the grating along a single bohmian trajectory through-passing one and only one slit. Keywords: Gaussian wavepacket, neutron scattering, guidance equation, bohmian trajectory, near-field interference, far-field diffraction, Talbot carpet, fractalComment: 12 pages, 14 figures; added one new section, 6 new figures, and 4 references; added 1 figure, some comments, renew 3 figures

Quantum consciousness in warm, wet, and noisy brain

The emergence of quantum consciousness stems from dynamic flows of hydrogen ions in brain liquid. This liquid contains vast areas of the fourth phase of water with hexagonal packing of its molecules, the so-called exclusion zone (EZ) of water. The hydrogen ion motion on such hexagonal lattices shows as the hopping of the ions forward and the holes (vacant places) backward, caused by the Grotthuss mechanism. By supporting this motion using external infrasound sources, one may achieve the appearance of the superfluid state of the EZ water. Flows of the hydrogen ions are described by the modified Navier-Stokes equation. It, along with the continuity equation, yields the nonlinear Schrodinger equation, which describes the quantum effects of these flows, such as the tunneling at long distances or the interference on gap junctions.Comment: 20 pages, 11 figure

Hydrodynamics of the Physical Vacuum: II. Vorticity dynamics

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 core where the orbital speed vanishes. The speed reaches a maximal value on the core wall and further it decreases monotonically. The vortex trembles around some average value and possesses by infinite life time. The vortex ball resulting from topological transformation of the vortex ring is considered as a model of a particle with spin. Anomalous magnetic moment of electron is computed.Comment: Revised e-print 1504.07497v1, 11 pages, 10 figures, (Foundations of Physics, in press

Hydrodynamics of Superfluid Quantum Space: de Broglie interpretation of the quantum mechanics

The ubiquitous ether coming from the ancient times up to middle of the twenty century is replaced by a superfluid quantum space. It represents by itself a Bose-Einstein condensate consisting of enormous amount of virtual particle-antiparticle pairs emerging and disappearing in an infinitely ongoing dance. Flowing of this medium in the non-relativistic limit is described by the modified Navier-Stokes equation along with the continuity equation. The first equation admits the splitting on to two coupled equations. They are the quantum Hamilton-Jacobi equation and the equation for vorticity. The quantum Hamilton-Jacoby equation paired with the continuity equation can be reduced to the \Schrodinger equation. These two equations representing the kernel of the Bohmian mechanics give finding bundle of the Bohmian trajectories. Whereas the vorticity equation gives solutions for vortices moving along such trajectories. As the result we come to the de Broglie's interpretation of quantum mechanics according to which there is a pilot-wave guiding the particle (in our case it is a vortex clot) from a source up to its detection along an optimal path that is the Bohmian trajectory.Comment: 19 pages, 7 figures, Quantum Studies: Mathematics and Foundations, 2017, available on URL: http://rdcu.be/un4

N-slit interference: Path integrals, Bohmian trajectories

Path integrals give a possibility to compute in details routes of particles from particle sources through slit gratings and further to detectors. The path integral for a particle passing through the Gaussian slit results in the Gaussian wavepacket. The wavepackets prepared on N slits and superposed together give rise to interference pattern in the near-field zone. It transforms to diffraction in the far-field zone represented by divergent principal rays, at that all rays are partitioned from each other by (N-2) subsidiary rays. The Bohmian trajectories in the near-field zone of N-slit gratings show wavy behavior. And they become straight in the far-field zone. The trajectories show zigzag behavior on the interference Talbot carpet (ratio of particle wavelength to a distance between slits are much smaller than 1 and N>>1). Interference from the the N-slit gratings is simulated by scattering monochromatic neutrons (wavelength=0.5 nm). Also we have considered simulation of interference fringes arising at scattering on an N-slit grating of fullerene molecules (according to the real experiment stated in e-print 1001.0468).Comment: 17 pages, 16 figures, added simulation of the fullerene molecular interference and 4 figures. PACS numbers: 03.75.-b, 03.75.Dg, 42.25.Fx, 42.25.Hz, 45.20.Jj, 47.10.Df, 61.05.f

From the Newton's laws to motions of the fluid and superfluid vacuum: vortex tubes, rings, and others

Owing to three conditions (namely: (a) the velocity is represented by sum of irrotational and solenoidal components; (b) the fluid is barotropic; (c) a bath with the fluid undergoes vertical vibrations) the Navier-Stokes equation admits reduction to the modified Hamilton-Jacobi equation. The modification term is the Bohmian(quantum) potential. This reduction opens possibility to define a complex-valued function, named the wave function, which is a solution of the Schr\"{o}dinger equation. The solenoidal component being added to the momentum operator poses itself as a vector potential by analogy with the magnetic vector potential. The vector potential is represented by the solenoidal velocity multiplied by mass of the fluid element. Vortex tubes, rings, and balls along with the wave function guiding these objects are solutions of this equation. Motion of the vortex balls along the Bohmian trajectories gives a model of droplets moving on the fluid surface. A peculiar fluid is the superfluid physical vacuum. It contains Bose particle-antiparticle pairs. Vortex lines presented by electron-positron pairs are main torque objects. Bundles of the vortex lines can transmit a torque from one rotating disk to other unmoved disk.Comment: 14 pages, 9 figure

Dark matter is a manifestation of the vacuum Bose-Einstein condensate

The vorticity equation stemming from the modified Navier-Stokes equation gives a solution for a flat profile of the orbital speed of spiral galaxies. Solutions disclose existence of the Gaussian vortex clouds, the coherent vortices with infinite life-time, what can be a manifestation of the dark matter. The solutions also disclose what we might call a breathing of the galaxies - due to an exchange of the vortex energy with zero-point fluctuations in the vacuum. .Comment: 7 pages, 5 figure

Hydrodynamics of the physical vacuum: dark matter is an illusion

The relativistic hydrodynamical equations are being examined with the aim of extracting the quantum-mechanical equations (the relativistic Klein-Gordon equation and the Schr\"odinger equation in the non-relativistic limit). In both cases it is required to get the quantum potential, which follows from pressure gradients within a superfluid vacuum medium. This special fluid, endowed with viscosity allows to describe emergence of the flat orbital speeds of spiral galaxies. The viscosity averaged on time vanishes, but its variance is different from zero. It is a function fluctuating about zero. Therefore the flattening is the result of the energy exchange of the torque with zero-point fluctuations of the physical vacuum on the ultra-low frequencies.Comment: 10 pages, 4 figure

Hydrodynamics of Superfluid Quantum Space: particle of spin-1/2 in a magnetic field

The modified Navier-Stokes equation describing the velocity field in the superfluid quantum space is loaded by the external Lorentz force introducing electromagnetic fields. In order to open the path for getting the \Schrodinger-Pauli equation describing the behavior of a particle with spin-1/2 in the magnetic field we need to extend the continuity equation to take into account conservation of spin flows on the 3D sphere. This extension includes conservation of the density distribution function in 6D space, that is a multiplication of the 3D Euclidean space by the 3D sphere of unit radius. The special unitary group SU(2) underlies the rotations of the spin on this sphere. This group is isomorphic to the group of quaternions containing the real 4x4 matrices of norm 1. Transition to the quaternion group opens up the way to the possibility of describing the spin-1/2 behavior in a magnetic field as a motion of a spin flag on the 2D sphere. Maxwell's electromagnetic field theory manifests itself in the quaternion group basis by the natural manner.Comment: 22 pages, 6 figure