Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation,
(−2mℏ2∇2+v)Ψ=EΨ,ℏ=2πh, or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression
H=−2mℏ2∇2+V; h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and
2m−ℏ2∇2 the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.
We study the quantization of a classical system of interacting particles obeying a recently proposed kinetic interaction principle (KIP) [G. Kaniadakis, Physica A 296, 405 (2001)]. The KIP fixes the expression of the Fokker-Planck equation describing the kinetic evolution of the system and imposes the form of its entropy. In the framework of canonical quantization, we introduce a class of nonlinear Schrödinger equations (NSEs) with complex nonlinearities, describing, in the mean-field approximation, a system of collectively interacting particles whose underlying kinetics is governed by the KIP. We derive the Ehrenfest relations and discuss the main constants of motion arising in this model. By means of a nonlinear gauge transformation of the third kind, it is shown that in the case of constant diffusion and linear drift, the class of NSEs obeying the KIP is gauge-equivalent to another class of NSEs containing purely real nonlinearities depending only on the field rho=|psi|(2) .
After unsuccessful arguments with N.Bohr about the incompleteness of the non-relativistic electron (NE) created by Schrödinger in wave quantum mechanics (WQM), Einstein said: " I am more and more inclined do think that extra-spatial structures are necessary for the further development of physics ". And the extra-spatial structures predicted by Einstein had to be constants, since only constants were independent of the space-time continuum. Therefore, it was necessary to find an answer to the question: " How can constants be used as extra-spatial structures for the further development of physics?" The answer this question arose naturally when we managed to find out that Planck’s constant and other constants associated with it are the proper corpuscular-wave (CW) quantities of microparticles. At the same time, the newly discovered extra-spatial CW quantities of microparticles, being constants, turned out to be extra-spatial structures in relation to the Euclidean, Minkowski and Klein-Gordon spaces. In addition it turned out that our newly discovered extra-spatial CW quantities of microparticles inside the space of Euclidean, Minkowski and Klein-Gordon appear in the form of corpuscular, mixed and wave quantities of microparticles that are generally known to this day. Moreover, the emergence within spatial corpuscular quantities from extra-spatial CW quantities made it possible to understand that the corpuscular quantities of Newton’s corpuscular classical mechanics (CCM), which were considered the principles of physics, are not in fact the principles of physics. It turned out that the WQM of a NE created by Schrödinger is an incomplete version of the corpuscular-wave mechanics (CWM) of a NE. No matter discover the extra-spatial predicted by Einstein necessary for the further development of physics in the form of extra-spatial proper CW quantities and the CWM of real objects of Nature.