The Connection between Bohmian Mechanics and Many-Particle Quantum Hydrodynamics

Klaus Renziehausen,Ingo Barth

doi:10.1007/s10701-020-00349-1

Klaus Renziehausen, Ingo Barth

Open Access

https://doi.org/10.1007/s10701-020-00349-1

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Abstract

Bohm developed the Bohmian mechanics (BM), in which the Schrödinger 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 in 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.

Highlights

In 1952, Bohm used preworks of Madelung [1, 2] to develop his Bohmian mechanics (BM) [3, 4]
As a result of this discussion, we found that there is both a many-particle continuity equation of manyparticle quantum hydrodynamics (MPQHD), a many-particle Ehrenfest equation of motion, and the many-particle quantum Cauchy equation for each individual particle sort and for the total particle ensemble, where all these differential equations only depend on one position vector, too
In the early 1950s [3, 4], Bohm used an ansatz to transform the Schrödinger equation which leads to the result that the Schrödinger equation can be split in two other differential equations: The first of these equations is the continuity equation of BM that is related to the conservation of particles, and the second of these equations is the Bohmian equation of motion being very similar to the Newtonian equation of motion except for the detail that in the Bohmian equation of motion an additional quantity appears which is called the quantum potential

Summary

Introduction

In 1952, Bohm used preworks of Madelung [1, 2] to develop his Bohmian mechanics (BM) [3, 4]. As intermediate results of this derivation, we will find the many-particle Ehrenfest equation of motion both for each individual particle sort and for the total particle ensemble, too Performing this calculation, we will have to do an averaging over the coordinates of all particles except one because—as already stated—the differential equations of BM depend on the complete set of particle coordinates, while the differential equations of MPQHD, which we want to derive, only depend on a single position vector. 3, the proof is given how we can use the continuity equation of BM and the Eulerian version of the Bohmian equation of motion as a starting point to derive the continuity equation of MPQHD and the quantum Cauchy equation both for each individual particle sort and for the total particle ensemble.

Basic Physics of BM for a Many‐Particle System

Connection between BM and MPQHD

Summary

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