Nonlocal Physics in the Wave Function Terminology

Shortcomings of the Boltzmann physical kinetics and the Schrödinger wave mechanics are considered. From the position of nonlocal physics, the Schrödinger equation is a local equation; this fact leads to the great shortcomings of the linear Schrödinger wave mechanics. Nonlocal nonlinear quantum mechanics is considered using the wave function terminology.

9) Nonlocal physical kinetics brings the strict approximation of non-local effects in space and time and after transfer to the local approximation leads to parameter τ , which on the quantum level corresponds to the uncertainty principle "time-energy". Methods of the τ definition in [2]- [10] are considered.
It is established that the theory of transport processes (including quantum mechanics) can be presented within the framework of the universal theory (unified theory of dissipative systems) based on the nonlocal physical description [2]- [10]. It is shown, in particular, that the equations of nonlocal physics lead to the appearance of solitons, which supports the Schrödinger opinion, who interpreted quantum mechanics from the point of view of the existence of waves of matter.
Let us turn to the logic of the development of the non-local theory of transport processes: 1) In 1926 Madelung published a brilliant article [11] in which he transformed the quantum postulate (Schrödinger equation containing the ψ wave function) in hydrodynamics. In other words, the evolution of a single bound electron was possible to interpret as some effective flow. The Schrödinger equation is not dissipative. Therefore, generalized quantum hydrodynamics is a tool for solving problems in the theory of dissipative nano-systems.
Here just note that: 1) SE is not able to give a self-consistent description of the nucleus-electron shell.
2) SE does not lead to an independent analogue of the hydrodynamic energy equation.
3) SE (and its equivalent hydrodynamic Madelung form) is not a dissipative equation and therefore cannot be applied to the description of dissipative processes in nanotechnology.
4) The Schrödinger equation cannot explain spontaneous emission, since the wave function of the excited state is an exact solution of the time-dependent SE equation [13].
5) The linear Schrödinger equation cannot describe the measurement process in quantum mechanics, since the measurement process is nonlinear, stochastic and irreversible in time.
6) The Schrödinger equation cannot describe the processes of mutual transformations of elementary particles.
7) The Schrödinger equation is not able to describe the "nucleus-electron shell" complex as a whole.
8) The Schrödinger equation is not able to describe the spatial electron shell without using additional assumptions, such as the Pauli principle. 9) To a large extent, the result of quantization is the result of cutting off infinite series and turning them into polynomials. This process resembles the transformation of a traveling wave into a system of standing waves if a reflection from an obstacle is introduced.
As you know, the basic equation of quantum mechanics-the Schrödinger equation is written in terms of the wave function and is, in fact, a postulate. The Schrödinger equation is "guessed" based on reasonable physical considerations. The main stages of similar "derivation" can be characterized as follows: 1) The desired equation should reflect the wave properties of particles, including one-dimensional harmonic oscillations ( ) Relation (1.1) includes the circular frequency ω and wave number where λ is the wavelength. In accordance with the de Broglie principle, we compare this oscillation to a certain corpuscular object and introduce the connection of its kinetic energy к E and momentum p with the frequency attributed to it: Let us differentiate the wave function (1.6) by time and multiply both parts of (1.7) by i when differentiating, it is assumed that the kinetic energy and momentum of the particle do not depend on the coordinates and time, and, therefore, we are talking about a free particle. Because it is possible to record (1.9) in the form of Compare now (1.8) and (1.11), the right parts of these relations are the same.
Therefore, recording is possible 4) The following reasonable generalization: a) if we consider 3D case: b) if we consider now the particle movement in a potential field ; this potential energy should be added on the right-hand side of (1.13): Journal of Applied Mathematics and Physics (1.14) Equation (1.14) was derived in 1926 by Erwin Schrödinger and bears his name.
Schrödinger's first message [14] is entitled "Quantization as an eigenvalue problem." Some comments to this equation: 1) Restriction on the mass of the particle is not entered. However, the particle is considered as a material point. Generalization of the equation to many-particle systems is a separate problem. To solve, for example, the problem of the evolution of a system containing n electrons, it is necessary to consider ψ as a function of 3n independent coordinates and time t.
2) The Schrödinger equation belongs to the class of linear equations. This means that in the sum of some solutions 1 ψ and 2 ψ of the Schrödinger equation there is also a solution to this equation 2) The Schrödinger equation does not describe dissipative processes.
3) The Schrödinger equation is not able to describe the whole complex "nucleus-electron shell".
4) The Schrödinger equation is unable to describe a spatial electron shell without the use of additional assumptions such as the Pauli principle.

5)
To a large extent, the quantization result is the result of cutting infinite series and turning them into polynomials.
The next obvious step was taken by E. Madelung (E. Madelung [11]) in 1926.
We are talking about the derivation of a special form of the Schrödinger equation after the wave function representation in the form of  (1.16) and the equation of motion for the Euler potential flow where the effective potential has the form Schrödinger assumed, that the spatial evolution of a quantum object (such as an electron) can be described as the motion of a wave packet, in other words, a soliton. But Pauli showed that the wave packet, built on Schrödinger, is spreading in space.
Really, the non-stationary 1D Schrödinger Equation (SE) can be writes as where ψ is the wave function. In the ID case we have The solution (1.23) has the form ( )

25) Journal of Applied Mathematics and Physics
where 0 k is the dimensionless wave number. We notice also that ( ) and comparing (1.29) and (1.28) we find identity (1.23).
Let us find now Using the De Moivre formula (1707) and geometric relations [15]) we obtain (1.33) Journal of Applied Mathematics and Physics As we see It is evident that this dispersive wave packet, while moving with constant group velocity ( 0 const k = ), is delocalizing rapidly (if t → ∞ ). This fact leads to dramatic consequences not only for the theoretical physics but for biology also (see [16]).
The nonlocal physics radically improve the situation; as a result we obtain sta-   , p,

Generalized Hydrodynamic Equations of Nonlocal Physics and Madelung Wave Function
The generalized hydrodynamic equations (GHE) can be obtained from the nonlocal kinetic equation in the frame of the Enskog procedure, (see for example [2] [3] [4] [5]). Generally speaking to GHE should be added the system of genera- Here ( )

About the Non-Local Hydrodynamic Equations in ψ-Interpretation
The following conclusions of fundamental importance can be made:  In NLQH we needn't to use the formalism of the ψ function description. Then we needn't finding the physical interpretation of ψ function.
From position of nonlocal physics ψ -function is only a mathematical function (between many others) defining by relations (2.8)-(2.10). The attempt to find any physical sense for ψ has no reason.
But physical community accustomed to "the ψ interpretation". Then it is reasonable to find "the ψ interpretation" of non-local hydrodynamic equa- Using the Madelung identifications The subsequent transformations are of a fundamentally important nature and we will give detailed calculations   (3.13) or Journal of Applied Mathematics and Physics  2) Omitting nonlinear terms in the generalized continuity equation but conserving the terms proportional to qu τ we obtain other known quantum equations. In particular, the second term in the left-hand side of Equation ( The physics of the twenty-first century is nonlocal physics.