Topological Solution of Bohmian Quantum Mechanics

Abstract

The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrödinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer degree. The evolution of the world line at the bifurcation point is given.

Knot soliton in DNA and geometric structure of its free-energy density

“…Here, we use the Einstein summation convention in (18), (20) and (22). Then the total free-energy density changes to:…”

“…By considering the ϕ-mapping topological current theory [20][21][22], the vorticities of the strands j 1 and j 2 are:…”

The Symmetry of Current of the Coherent State from the View of O(3) σ-Model

“…In traditional quantum mechanics, the curl of the velocity vanishes at zero points of the wave functions. However, the curl of the velocity must be modified along trajectories because ∇ × V need not vanish at nodal points of the wave function [18]. The curl of the velocity is…”

“…The topology and geometry play an important role in physics and mathematics and a great deal of works have been done in the topology and geometry [12][13][14][15][16][17]. Especially, the vorticity of the vortex in condensate meter and topology of the physical system have been studied by applying the φ−mapping topological current theory [18][19][20][21]. In this paper, we present the relation between the current of coherent state and the supercurrent of the two-gap condensate system.…”

Different electronic charges in two-component superconductor by coherent state

“…The ratios of the different charges are studied in the coherent state and incoherent state and we find the ratios are different. Based on φ-mapping topological theory [15][16][17], we discuss which condition leads to integer vorticity and which condition leads to fractional vorticity. The contents is arranged as follows: in Section 2, we suggest the superpositions describing the two condensates are coherent.…”