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arXiv:quant-ph/9911007 · Physical Review A

Motion of vortex lines in quantum mechanics

Iwo Białynicki‐Birula, Zofia Białynicka-Birula, Cezary Śliwa

Published Feb 16, 2000• 134 citations
arXivHTMLPDF

Abstract

Exact analytic solutions of the time-dependent Schr\"odinger 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.

DOI
10.1103/physreva.61.032110

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Exact analytic solutions of the time-dependent Schr"odinger 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.
In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrödinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrödinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines. Although the equation of motion of the quantum mechanical probability fluid is different in its literal form from the equations describing the real physical fluid, we believe that the evolution of the vorticity in the quantum and in the real fluid share the same qualitative features that can be described in terms of the topology of the vortex lines configurations. The general phenomena such as the switch-over, creation and annihilation of vortices can be observed in the quantum mechanical fluid. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A family of exact analytical solutions of vortices in quantum fluid governed by a two-dimensional time-dependent Schrödinger equation is presented, which describes different kinds of vortex structures. The dynamics of different vortex clusters, such as the single vortex, vortex pair, vortex dipole and vortex trimer in a two-dimensional quantum fluid are analytically studied based on these exact solutions. The time evolutions of the wave of such vortices are demonstrated, and the orbits of motion of singular points in the vortices are also explored. The interactions of vortices in many-vortex clusters are discussed. A repulsive interaction between vortices with the same topological charge, and inter-annihilation and inter-creation of vortices with opposite topological charge, are shown.
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 addition, the 3D vortex line-line correlation exhibits universal behaviour, scaled as \Deltar^-2, where \Deltar is the separation between any two vortex line elements, in fully developed turbulence. We also show that the quantum vortex is not frozen to the matter, nor is the vortex motion induced by other vortices via Biot-Savart's law. Thus, the quantum vortex is actually a nonlinear wave, propagating at a speed very different from a classical vortex.
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Starting with a microscopic Hamiltonian, we derive the Schr\"odinger equation of a superconductor with a vortex. The analysis describes the many-body state of a superconductor as a delocalized superposition of Bogoliubov-de Gennes mean-field vortex states. Taking a suitable one-band extended Hubbard model as the microscopic Hamiltonian of electrons in high-Tc{T}_{c}Tc​ compounds, solutions to the vortex Schr\"odinger equation are computed. We compute the vortex mass. Tuning the Hamiltonian parameters, we obtain solutions for the case of an sss-wave order parameter and for the ddd-wave case and consider consequences of order parameter symmetry for vortex directional motion.