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Acta Polytechnica

Non Mechanical (Mezic) Type Forces in the Foundations of Quantum Mechanics

Čestmír Šimáně

Published Jan 6, 2010• 0 citations
Publisher

Abstract

Many authors have attempted to derive the fundamental equations of quantum mechanics from classical hydrodynamics. In the present contribution we presume that the continuous, electrically charged material substance moves simultaneously under the influence of the electric field and at the same time undergoes a diffusion process. This assumption leads to the appearance of non-mechanical (mezic) type forces responsible for inner sources of matter (positive or negative), similar to those whose existence is supposed to exist in relativistic hydrodynamics. We obtained a non-linear differential equation, convertible by linearization to a form coinciding with the Schrödinger equation, as a condition for the establishment of the same steady states with discrete energies.

DOI
10.14311/1301

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Many authors have attempted to derive the fundamental equations of quantum mechanics from classical hydrodynamics. In the present contribution we presume that the continuous, electrically charged material substance moves simultaneously under the influence of the electric field and at the same time undergoes a diffusion process. This assumption leads to the appearance of non-mechanical (mezic) type forces responsible for inner sources of matter (positive or negative), similar to those whose existence is supposed to exist in relativistic hydrodynamics. We obtained a non-linear differential equation, convertible by linearization to a form coinciding with the Schrödinger equation, as a condition for the establishment of the same steady states with discrete energies.
Relativistic equations were derived of the motion of a continuous, electrically and mesically charged substance obeying the simple diffusion law and moving in a constant external electrical scalar potential. The relevant equation of continuity is presupposed with the right side different from zero, corresponding to the internal sources of matter. It is shown that if the matter distribution satisfies this equation of continuity, the gradient of the external potential is just compensated by the inner potential gradient of the electrically charged matter. In this case, the acceleration of the substance is zero and all dynamical variables are time independent. Thus, the validity of the equation of continuity has to be considered as a necessary and sufficient condition for the existence of steady, time independent distribution of the continuous substance density. Our considerations are limited to continua with spins equal zero because the osmotic velocity satisfies the simple diffusion law. The simplest way to find a solution of the nonlinear equation of continuity is its linearization, leading directly to the Schrodinger equation for steady bound states, from which the eigenvalues can be obtained. Substituting the latter into the nonlinear equation of continuity, one can reveal its solution.
By introducing in the Euler equation non mechanical (mesic) force densities equal to the products of corresponding potentials and gradient of the matter density, and supposing that the dynamical process has a diffusion character, the equation of continuity with internal sources (both positive and negative) is obtained. These sources are proportional to the algebraical sum of electrostatic, kinetic and proper energy densities, the constant of proportionality being the inverse value of the diffusion constant D = h/2m, taken with negative sign. The stationary state is obtained if the solution of the continuity equation holds in any point of the object, which is possible only for discrete values of the constant E with dimension of energy, entering in the equation of continuity. The equation of continuity can be transformed to the Bohm equation with the quantum potential, which may be solved by reducing it to the corresponding Schrodinger equation. Thus the spatial Concepts of Physics, Vol. V, No. 3 (2008) DOI: 10.2478/v10005-007-0043-6 499 distribution of the density of matter is given by the square of the Schrodinger wave function. Once the density distribution function is known, the velocity ~u in the diffusion process can be calculated and the process is deterministic. Superimposed to the diffusion process is the classical motion with velocity ~v, for which the equation of continuity without internal sources of matter holds. 500 Concepts of Physics, Vol. V, No. 3 (2008) Discrete states of continuous electrically charged matter
A new theoretical concept of quantum mechanics is proposed. The leading role is assigned to the electron as a non-point particle with a volume density of charge and mass. Based on the Hamilton-Jacobi equation, a nonlinear differential equation describing the dynamics of the charged substance of an electron is proposed. This new fundamental equation is transformed into the Schrödinger equation, with the density of the material substance of the electron being proportional to the square of the wave function. Since an electron in the form of a "cloud" of matter can change its configuration in space, we can give a classical interpretation to the process of generating a photon with a frequency and energy corresponding to the principles of quantum mechanics. Interference, diffraction, and the non-relativistic concept of electron spin is discussed.
Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time-dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.
The nonrelativistic quantum theory of a particle having both electric and magnetic charges moving in an arbitrary external electromagnetic field is presented. The theory is based on the hydrodynamic formulation of quantum mechanics. Dirac's quantization condition for the electric and magnetic charges is rederived as a consistency condition for the motion of the probability fluid. Neither the wave function nor the electromagnetic potential, which were the source of ambiguities in all other formulations, appears in our approach. Nevertheless, this theory has all the essential features of the standard quantum mechanics, including the superposition principle.
The theory of non-dissipative diffusion is constructed on an example of diffusion of a light particle in a dilute medium of heavy particles and it is shown that in low dissipative systems there are specific effects of non-dissipativity similar to quantum effects.In a non-dissipativity region mean energy of the light particle is conserved and processes are described by two non-linear diffusion equations with forward and backward time derivatives.Then these two diffusion equations are linearized and give one linear Schrödinger equation for complex amplitudes of probability.As a result, in the non-dissipative classical diffusion should be added probability amplitudes and there holds the superposition principle for these amplitudes.A mean square length of free passage and a mean square momentum define an elementary phase volume and a diffusion constant and they obey the uncertainty relations.It is shown that the formalism of quantum mechanics describes the classical non-dissipative diffusion with a homogeneous diffusion constant and that quantum mechanics is only a particular case when the elementary phase volume of free passage is universal and equal to the Planck constant.