The connection between Bohmian mechanics and many-particle quantum hydrodynamics

TL;DR

This paper demonstrates how many-particle quantum hydrodynamics equations can be derived from Bohmian mechanics and argues that MPQHD equations are more practical for analyzing many-particle systems due to their dependence on a single position vector.

Contribution

It shows the derivation of MPQHD differential equations from Bohmian mechanics for multi-particle ensembles with different particle types.

Findings

MPQHD equations depend on a single position vector, unlike BM equations.

MPQHD is more suitable for many-particle system analysis.

The derivation clarifies the connection between BM and MPQHD.

Abstract

Bohm developed the Bohmian mechanics (BM), in which the Schr\"odinger equation is transformed into two differential equations: A continuity equation and an equation of motion similar to the Newtonian equation of motion. This transformation can be executed both for single-particle systems and for many-particle systems. Later, Kuzmenkov and Maksimov used basic quantum mechanics for the derivation of many-particle quantum hydrodynamics (MPQHD) including one differential equation for the mass balance and two differential equations for the momentum balance, and we extended their analysis in a prework [K. Renziehausen, I. Barth, Prog. Theor. Exp. Phys. 2018, 013A05 (2018)] for the case that the particle ensemble consists of different particle sorts. The purpose of this paper is to show how the differential equations of MPQHD can be derived for such a particle ensemble with the differential equations of BM as a starting point. Moreover, our discussion clarifies that the differential equations of MPQHD are more suitable for an analysis of many-particle systems than the differential equations of BM because the differential equations of MPQHD depend on a single position vector only while the differential equations of BM depend on the complete set of all particle coordinates.