Differential equations in quantum mechanics

акустика Механика

Klaus Renziehausen, Ingo Barth

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.

Vikke Vuorenpää

Dmitri Yerchuck, Alla Dovlatova, Andrey F. Alexandrov

The symmetry study of main differential equations of mechanics and electrodynamics has shown, that differential equations, which are invariant under transformations of groups, which are symmetry groups of mathematical numbers (considered in the frame of the number theory) determine the mathematical nature of the quantities, incoming in given equations. It allowed to proof the main postulate of quantum mechanics, that to any mechanical quantity can be set up into the correspondence the Hermitian matrix by quantization. High symmetry of Maxwell equations allows to show, that to EM-field funcions, incoming in given equations, can be set up into the correspondence the Quaternion (twice-Hermitian) matrices by their quantization.

Action at distance in Newtonian physics is replaced by finite propagation speeds in classical post-Newtonian physics.As a result, the differential equations of motion in Newtonian physics are replaced by functional differential equations, where the delay associated with the finite propagation speed is taken into account.Newtonian equations of motion, with post-Newtonian corrections, are often used to approximate the functional differential equations.Are the finite propagation speeds the origin of the quantum mechanics?In this work a simple atomic model based on a functional differential equation which reproduces the quantized Bohr atomic model is presented.As straightforward application of the result the fine structure of the hydrogen atom is approached.

G. Fano, S. M. Blinder