The connection between Bohmian mechanics and many-particle quantum hydrodynamics

Bohm developed an ansatz to transform the Schr\"odinger equation into two differential equations. One of them is a continuity equation related to the conservation of the number of particles, and the other is an equation of motion similar to the Newtonian equation of motion. This ansatz for the analysis of quantum mechanics is called Bohmian mechanics (BM). These two differential equations of BM can be derived both for single-particle systems and for many-particle systems and depend on the complete set of particle coordinates of the system. Later, Kuzmenkov and Maksimov used basic quantum mechanics for the derivation of many-particle quantum hydrodynamics (MPQHD) including the derivation of three differential equations: One equation for the mass balance and two differential equations for the momentum balance. In a prework [K. Renziehausen, I. Barth, Prog. Theor. Exp. Phys. 2018, 013A05 (2018)], we extended this analysis by the case that the particle ensemble consists of different sorts of particles. For such a system, a version for each of the above-mentioned three differential equations of MPQHD can be found both for each individual particle sort and for the total particle ensemble. All these differential equations of MPQHD only depend on a single position vector. The purpose of this paper is to show how the equations mentioned above, which are related to either BM or MPQHD, are connected with each other -- therefore, we prove how all the above-mentioned differential equations of MPQHD can be derived when we use the two above-mentioned 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 they depend only a single position vector and not on the complete set of particle coordinates of the system.