Nonlinear Riccati equations as a unifying link between linear quantum mechanics and other fields of physics

Searching for the first well-documented attempts of introducing some kind of "quantization" into the description of nature inevitably leads to the ancient Greeks, in particular Plato and Pythagoras. The question of finding the so-called Pythagorean triples, i.e., right-angled triangles with integer length of all three sides, is, surprisingly, connected with complex nonlinear Riccati equations that occur in time-dependent quantum mechanics. The complex Riccati equation together with the usual Newtonian equation of the system, leads to a dynamical invariant with the dimension of an action. The relation between this invariant and a conserved "angular momentum" for the motion in the complex plane will be determined. The "Pythagorean quantization" shows similarities with the quantum Hall effect and leads to an interpretation of Sommerfeld's fine structure constant that involves another quantum of action, the "least Coulombic action" e2/c. Since natural evolution is characterized by irreversibility and dissipation, the question of how these aspects can be incorporated into a quantum mechanical description arises. Two effective approaches that also both possess a dynamical invariant (like the one mentioned above) will be discussed. One uses an explicitly time-dependent (linear) Hamiltonian, whereas the other leads to a nonlinear Schrödinger equation with complex logarithmic nonlinearity. Both approaches can be transformed into each other via a non-unitary transformation that involves Schrödinger's original definition of a (complex) action via the wave function.

Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian

The basis of modern conception of the world consists of two phenomenological theories (theory of quantum mechanics and theory of relativity), both largely inconsistent, but, in a number of cases, suitable for evaluation of experimental data. Both of these theories have one thing in common: their authors are convicted in limitations of laws and equations of classical mechanics, in absolute validity of Maxwell equations and in essential distinction of laws and mechanisms of the device of macrocosm and microcosm. Nevertheless, such assurance, being dominant in physics in the last hundred years, hasn’t resulted in creation of unifying fundamental physical theory, nor in essential understanding of principal physical conceptions, such as: electric, magnetic and gravitational fields, matter and antimatter, velocity of light, electron, photon and other elementary particles, internal energy, mass, charge, spin, quantum properties, Planck constant, fine structure constant and many others. All laws and the equations of modern physics are attempts to approximate description of the results of natural experiments, rather than strict theoretical (mathematical) findings from the general and uniform laws and mechanisms of the device of the world surrounding us. Moreover, some conclusions from modern physics equations contradict experimental data such as infinite energy or mass of point charge. In papers (Magnitskii, 2010a, 2011a) bases of the unifying fundamental physical theory which a single postulate is the postulate on existence of physical vacuum (ether) are briefly stated. It is shown, that all basic equations of classical electrodynamics, quantum mechanics and gravitation theory can be derived from two nonlinear equations, which define dynamics of physical vacuum in three-dimensional Euclidean space and, in turn, are derived from equations of Newtonian mechanics. Furthermore, clear and sane definitions are given to all principal physical conceptions from above through the parameters of physical vacuum, namely its density and propagation velocity of various density’s perturbations. Thereby, it is shown that a set of generally unrelated geometric, algebraic and stochastic linear theories of modern physics, which are fudged to agree with experimental data and operating with concepts of multidimensional spaces and space-time continuums, can be replaced with one nonlinear theory of physical vacuum in ordinary three-dimensional Euclidean space, based exclusively on laws of classical mechanics. In the present paper research of system of equations of physical vacuum is continued with the purpose of studying and the description of processes of a birth of elementary particles and their properties. A system of equations of electrodynamics of the physical vacuum,

The factorisation method commonly used in linear supersymmetric quantum mechanics is extended, such that it can be applied to nonlinear quantum mechanical systems. The new method is distinguishable from the linear formalism, as the superpotential is forced to become eigenfunction-dependent. Example solutions are given for the nonlinear Schrödinger equation and its supersymmetric partner equation. This method allows new nonlinear evolution equations to be constructed from the solutions of known nonlinear equations, and has the potential to be a useful tool for mathematicians and physicists working in the field of nonlinear systems, allowing the discovery of previously unknown ‘dualities’ amongst nonlinear evolution equations.

It is shown that a nonlinear reformulation of time-dependent and time-independent quantum mechanics in terms of Riccati equations not only provides additional information about the physical system, but also allows for formal comparison with other nonlinear theories. This is demonstrated for the nonlinear Burgers and Korteweg–de Vries equations with soliton solutions. As Riccati equations can be linearized to corresponding Schrödinger equations, this also applies to the Riccati equations that can be obtained by integrating the nonlinear soliton equations, resulting in a time-independent Schrödinger equation with Rosen–Morse potential and its supersymmetric partner. Because both soliton equations lead to the same Riccati equation, relations between the Burgers and Korteweg–de Vries equations can be established. Finally, a connection with the inverse scattering method is mentioned.

We discuss a model of non linear quantum mechanics in which the wave equation satisfies the homogeneity condition (2.1). It is argued that in this model the set of (mixed) states is a simplex.

Quantum mechanics is the most ground-breaking and fascinating theoretical concept developed in physics during the past century. Much of our present understanding of the microscopic world and its extension into the macroscopic world, including modern technical applications, is based upon quantum mechanics. We have experienced a remarkable development of information and communication technology during the past two decades, to a large extent depending upon successful fabrication of smaller and smaller components and circuits. However, we are finally approaching the physical limits of component miniaturization as we enter a microscopic world ruled by quantum mechanics.

In this Letter, a theorem on time-dependent linear Hamiltonian systems is recalled and its connection with the Schrodinger equation is discussed. The kernel of the evolution operator of such quantum systems is computed. Furthermore, the Lewis and Riesenfeld theory for systems with many degrees of freedom is generalized.

Complex adaptive learning is intelligent.It is adaptive, learns in feedback loops, and generates hidden patterns as many individuals, elements or particles interact in complex adaptive systems (CAS).CAS highlights adaptation in life and lifeless complex systems cutting across all traditional natural and social sciences disciplines.However, discovering a universal law in CAS and understanding the underlying mechanism of distribution formation, such as a non-Gauss distribution in complex quantum entanglement, remains highly challenging.Quantifying the uncertainty of CAS by probability waves, the authors explore the inherent logical relationship between Schrdinger's wave equation in quantum mechanics and Shi's trading volume-price wave equation in finance.Subsequently, the authors propose a non-localized wave equation in quantum mechanics if cumulative observable in a time interval represents momentum or momentum force in Skinner-Shi (reinforcement-frequency-interaction) coordinates.It supports the assumption that the invariance of interaction as a universal law exists in quantum mechanics and finance.The theory shows that quantum entanglement or adaptively interactive coherence is an interactively coherent state instead of a consequence of the superposition of coherent states.As a resource, quantum entanglement is non-separable, steerable, and energy-consumed.The entanglement state has opposite states subject to interaction conservation between the momentum and reversal forces.

If a wave function is written in polar form it becomes possible to write the Schrödinger equation of nonrelativistic quantum mechanics in a form analogous to the classical Hamilton-Jacobi equation with an extra term known as the quantum potential. Time-dependent supersymmetry is a procedure for finding new solutions of the Schrödinger equation if one solution is known. In this paper a time-dependent supersymmetry transformation is applied to a wave function in this polar form and it is shown that the classical potential plus the quantum potential is a conserved quantity under this transformation under certain circumstances. This leads to a modification of our view of the role of the quantum potential and also to a deeper appreciation of the function of a supersymmetry transformation.

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integrals of motion. We construct the most general cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in Cartesian coordinates with a third order integral are known. The general formalism is applied to quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E2. We also discuss these potentials from the point of view of supersymmetric and PT-symmetric quantum mechanics.

We revisit the localization computation of the expectation values of ’t Hooft operators in mathcal{N} = 2* SU(N) theory on ℝ3 × S1. We show that the part of the answer arising from “monopole bubbling” on ℝ3 can be understood as an equivariant integral over a Kronheimer-Nakajima moduli space of instantons on an orbifold of ℂ2. It can also be described as a Witten index of a certain supersymmetric quiver quantum mechanics with mathcal{N} = (4, 4) supersymmetry. The map between the defect data and the quiver quantum mechanics is worked out for all values of N. For the SU(2) theory, we compute several examples of these line defect expectation values using the Witten index formula and confirm that the expressions agree with the formula derived by Okuda, Ito and Taki [16]. In addition, we present a Type IIB construction — involving D1-D3-NS5-branes — for monopole bubbling in mathcal{N} = 2* SU(N) SYM and demonstrate how the quiver quantum mechanics arises in this brane picture.

We study entanglement of spin degrees of freedom with continuous one in supersymmetric (SUSY) quantum mechanics. Concurrence is determined by mean value of spin and is calculated explicitly for SUSY states. We show that eigenstates of supercharges are maximally entangled. As an example the entanglement of atom state with photon state and SUSY in Jaynes–Cummings model are considered.

The aim of this research is to solve the Hamilton–Jacobi–Bellman equation (HJB) arising in nonlinear optimal problem using Adomian decomposition method. First Riccati equation with matrix variable coefficients, arising in linear optimal and robust control approach, is considered. By using the Adomian method, we consider an analytical approximation of the solution of nonlinear differential Riccati equation. An application in optimal control is presented. The solution in different order of approximations and different methods of approximation will be compared with respect to accuracy. Then the HJB equation, obtained in nonlinear optimal approach, is considered and an analytical approximation of the solution of it, using Adomian method, is presented.

Ontic structural realists draw support for their view from fundamental theories in modern physics. Quantum mechanics is uncontroversially regarded as being amongst those theories. In this paper, I defend the idea that quantum mechanics cannot play this role of a fundamental theory without assuming that a solution for the measurement problem is available. I go on to illustrate that, in the light of possible solutions to the measurement problem, the support for ontic structural realism from quantum mechanics becomes disputable. I also argue that resorting to notions of fundamentality that leave the support for ontic structural realism intact presuppose too much aspects of this metaphysical position for the support from quantum mechanics to be considered significant.