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arXiv:1605.01922

An analogous Wu-Yang monopole in superfluid

Wei Jia, Yi-Rong Ma, Fang-Qi Hu, Qing Xiang Zhao

Published Mar 16, 2016• 0 citations
arXivHTMLPDF

Abstract

By identifying the Schrödinger equation with the hydrodynamic equations in superfluid ${^3}$He, the effective potential is introduced in the Schrödinger equation to solve the quantum pressure in steady state. The pure gauge velocity solutions of hydrodynamic equations provide an analogous Wu-Yang monopole potential. The two cases of velocity in superfluid are equivalent to the two regions of Wu-Yang monopole potential. Due to the compressibility of superfluid, the physical models are limited, such as hard core and harmonic oscillator. It is important that the constraint condition of $\nablaρ\cdot \textbf{v}=0$ plays a key role, which determines that the integral quantum numbers are selected in monopole harmonics and shows the special analogous Wu-Yang monopole. The results provide a new possibility for the simulation of Wu-Yang monopole.

DOI
10.48550/arxiv.1605.01922

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In the framework of Chronodynamic Cosmology (CDC), the vacuum is defined as a physical superfluid (Psi). If this premise is correct, the fundamental equations of Quantum Mechanics must be derivable from Hydrodynamics. In this note, we perform the Madelung Transformation to demonstrate that the Schrödinger Equation is mathematically equivalent to the Euler Equation for an inviscid, compressible fluid subjected to a specific internal pressure potential. This derivation suggests that "Quantum Potential" (Q) is the hydrostatic pressure of the Time-Fluid, and that "Wavefunction Collapse" is a phase transition from an inviscid (superfluid) state to a viscous (classical) state.
A nonlinear interaction term is added to the Schr\"odinger equation which makes it exactly equivalent to classical mechanics. In the hydrodynamic formulation of this equation, the usual classical theory is obtained. This theory is a Lagrangian density formulation of hydrodynamics and is used to calculate the density profile of superfluid helium in a line vortex. The theory can be canonically quantized, in the sense of quantum field theory, in terms of the density and velocity fields. The quantized theory is not equivalent to that which one would obtain if the quantization were done in the usual way in terms of the field operations ${\ensuremath{\psi}}^{*}$ and ($\ensuremath{\psi}$). The Schr\"odinger equation, with this interaction, has localized solutions with a definite energy and momentum and does not give quantized energy levels for a one-dimensional harmonic oscillator.
The theory of the condensate of a weakly interacting Bose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self-consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self-consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a boundary. This condition fully determines the dynamics of the system. These points are illustrated by considering the motion of a foreign ion in a Bose gas, a rotating container of fluid, and the Landau criterion for superfluidity.
This paper develops a hydrodynamic derivation of nonrelativistic quantum mechanics within the Hydrodynamic Quantum Gravity (HQG) programme. Using the Madelung transformation, the Schrödinger equation is written in amplitude–phase form and shown to be exactly equivalent to (i) a continuity equation for a conserved density and (ii) an Euler-type momentum equation augmented by the “quantum potential” term. In this interpretation, the quantum potential is treated as an effective dispersive stress associated with structured vacuum dynamics, rather than as a purely formal mathematical artefact. The paper reviews how standard quantum features appear naturally in the hydrodynamic variables: interference and diffraction as phase-driven flow, stationary states as time-harmonic solutions, and quantisation as a consequence of single-valued phase (circulation) constraints. It also discusses how pilot-wave style dynamics can be expressed in this language, and surveys laboratory analogues (including walking-droplet experiments) as phenomenological motivation and intuition-building, without claiming identity between the analogue systems and the quantum vacuum. Interpretation note: “Superfluid/condensate” language is used here as an effective mathematical/analogue description of vacuum behaviour; the paper does not assert a material substrate or an operationally accessible preferred rest frame. This paper is part of a series. Companion preprints develop the gravitational and electromagnetic components of the HQG framework and discuss additional open problems and testable predictions. Discussion and feedback: https://github.com/Gptham123456/Hydrodynamic-Quantum-Gravity/discussions
We introduce a nonlinear Schr\"odinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where $a{n}^{1/3}\ensuremath{\ll}1$ with aaa the interatomic s\mathit{s}s-wave scattering length and nnn the bosonic density, to the unitarity limit, where $a\ensuremath{\rightarrow}+\ensuremath{\infty}$. We call this equation the unitarity Schr\"odinger equation (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core radius in a uniform Bose gas. We also consider quantized vortices in a Bose gas under cylindrically symmetric harmonic confinement and study their profile and chemical potential using the USE and compare the results with those obtained from the Gross-Pitaevskii-type equations valid in the weak-coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose gas as a function of the scattering length.