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
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
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.
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.
We first recall that the system of fluid mechanics equations (Euler and continuity) that describes a fluid in irrotational motion subjected to a generalized quantum potential (depending on a constant which may be different from the standard quantum constant ℏ) is equivalent to a generalized Schrödinger equation for a 'wavefunction' whose modulus squared yields the fluid density. Then we show that even in the case of the presence of vorticity, it is also possible to obtain a nonlinear Schrödinger-like equation (now of the vectorial field type) from the continuity and Euler equations including a quantum potential. The same kind of transformation also applies to a classical charged fluid subjected to an electromagnetic field and to an additional potential having the form of a quantum potential. Such a fluid can therefore be described by an equation of the Ginzburg–Landau type, and is expected to show some superconducting-like properties. Moreover, a Schrödinger form can be obtained for a fluctuating rotational motion of a solid. In this case the mass is replaced by the tensor of inertia, and a generalized form of the quantum potential is derived. We finally reconsider the case of a standard diffusion process, and we show that, after a change of variable, the diffusion equation can also be given the form of a continuity and Euler system including an additional potential energy. Since this potential is exactly the opposite of a quantum potential, the quantum behavior may be considered, in this context, as an anti-diffusion.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
In a quantum bound state,spatial density is given by W(x)W(x), where W(x) is the wavefunction, average momentum by -i d ln(W) /dx and average kinetic energy at x by -1/2m d/dx d/dx W / W. One may note that: -1/2m d/dx d/dx W / W + V(x) = E is the time independent Schrodinger equation which may be solved for W(x) and E without any concept of spatial density. Furthermore, one may calculate and <p*p>, also without any concept of spatial density. This differs from the case of a classical fluid situation for which there is no W(x). We argue that kinematics of a bound state are contained in W(x) and that the Schrodinger equation is like a probability conservation hydrodynamic equation with spatial density replaced with W(x) and a V(x)W(x) term which we try to explain. In addition, the “conservation of momentum” hydrodynamic equation is essentially “d/dx” of the Schrodinger equation (if mass=m=.5). This strategy differs from the general approach used throughout the literature (for example (1) and many other cases), where spatial density is taken as the starting point of a quantum hydrodynamic formalism. Spatial density is utilized in a continuity equation of the form d/dt [W*W] + d/dx [W*W v] = 0 where v is a collective velocity and d/dt the partial derivative with respect to time. We argue one only needs the concept of a wavefunction as the normalization constant of the conditional probability P(p/x)= a(p)exp(ipx) / W(x) i.e. W(x)= sum over p a(p) exp(ipx). From this wavefunction, one may calculate and <p*p>. Thus we argue that hydrodynamic type equations should apply to W(x). We pointed out in (2), that W(x) in fact satisfies a momentum continuity hydrodynamic equation if mass=.5.
Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β2=ℏ/2m. In this article, we explore the implications of this view. Rewriting the wave function in a polar form and transforming Schrödinger's equation into two real equations, we show that one of them reduces to the continuity equation, and the other, a nonlinear dynamical equation for the probability density. Considering these two equations as if they were the basic equations of quantum mechanics, we apply them to several one-dimensional quantum systems. We show the dispersive properties in the probability densities of stationary states of a particle in a rigid box and in harmonic potential; quasi-classical Gaussian probability densities of a free particle; and coherent states and squeezed states of the harmonic oscillator. We also present the soliton as a quantum mechanical representation of a free particle. We discuss the meaning of the diffusion coefficient β2 for each quantum system using a density plot.