The Connection between Bohmian Mechanics and Many-Particle Quantum Hydrodynamics

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 manyparticle 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.

Developments of Bohmian Mechanics

“…Hence, Ψ = Ψ(r, t), where r = r 1 , • • • r n , i.e., we can consider Ψ to be a function of the spatial coordinates and time only. For Bohmian mechanics, [1][2][3][4][5][6][7][8][9][10][11][12][13] the wavefunction ansatz…”

“…Bohmian mechanics [1][2][3][4][5][6][7][8][9][10][11][12][13] is a deterministic theory of quantum mechanics that is based on a set of n velocity functions for n particles, where these functions depend on the wavefunction from the n-body time-dependent Schrödinger equation. The two equations of Bohmian mechanics are equivalent to the time-dependent Schrödinger equation.…”

A formulation of quantum fluid mechanics and trajectories

“…The energy equation of section 7 is a combination and generalization of the energy equations of the Bernoulli equation of fluid mechanics, used to describes 1-body states with a real valued wavefunctions [1], and the one from Bohmian Mechanics [2][3][4][5][6][7][8][9][10][11][12][13], used to describe states with complex value wavefunctions. The energy equation can be represented with a certain pressure, for fluids, or a certain body potential, for trajectories.…”

“…There are a number of overlapping formalism that go by various names for the description of quantum states: the De Broglie-Bohm theory [2][3][4][5][6][7][8][9][10][11][12][13], also called Bohmian mechanics and quantum hydrodynamics, and Madelung fluid mechanics [14,15]. All of these methods have the same Madelung velocity v, and, except for the earliest ones, explicitly contain the quantum potential Q [2,3,8].…”

Emergent gravity at all scales

“…The Bohmian interpretation of quantum mechanics, albeit not the most popular one, is perfectly equivalent to the standard ones, offers a novel and useful picture in many situations, such as the current one and elsewhere. 8,9 In other words, all predictions from the (Bohmian) approach are producible from the standard formalism of quantum mechanics, albeit in a slightly different way. The generalization of the above method to relativistic scenarios raises some interesting interpretational issues regarding the quantal trajectories, but the procedure as such is straightforward and once again yields identical predictions as from the conventional approach.…”