On the Dynamical Nature of Nonlinear Coupling of Logarithmic Quantum Wave Equation, Everett-Hirschman Entropy and Temperature

We study the dynamical behavior of the nonlinear coupling of a logarithmic quantum wave equation. Using the statistical mechanical arguments for a large class of many-body systems, this coupling is shown to be related to temperature, which is a thermodynamic conjugate to the Everett-Hirschman’s quantum information entropy. A combined quantum-mechanical and field-theoretical model is proposed, which leads to a logarithmic equation with variable nonlinear coupling. We study its properties and present arguments regarding its nature and interpretation, including the connection to Landauer’s principle. We also demonstrate that our model is able to describe linear quantum-mechanical systems with shape-changing external potentials.

Virtual beams and the Klein paradox for the Klein-Gordon equation

Whenever we consider any relativistic quantum wave equation we are confronted with the Klein paradox, which asserts that incident particles will suffer a surplus of reflection when dispersed by a discontinuous potential. Following recent results on the Dirac equation, we propose a solution to this paradox for the Klein-Gordon case by introducing virtual beams in a natural well-posed generalization of the method of images in the theory of partial differential equations. Thus, our solution considers a global reflection coefficient obtained from the two contributions, the reflected particles plus the incident virtual particles. Despite its simplicity, this method allows a reasonable understanding of the paradox within the context of the quantum relativistic theory of particles (according to the original setup for the Klein paradox) and without resorting to any quantum field theoretic issues.

APPLICATION DIRECT METHOD CALCULUS OF VARIATION FOR KLEIN-GORDON FIELD

Klein-Gordon field is often used to study the dynamics of elementary particles. The Klein–Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. The equation was found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron's spin, the equation failed to predict the fine structure of the hydrogen atom, and overestimated the overall magnitude of the splitting pattern energy. This paper will describe in detail using the Direct Method of Calculus Variation as an alternative to solve the Klien-Gordon field equations. The Direct Method simplified the calculation because the variables are calculated and expressed in functional form of energy. The result of the calculation of Klien-Gordon Feld provided the existence of the minimizer, i.e. with and . Explicit form of the minimizer was calculated by the Ritz method through rows of convergent density

SCHRÖDINGER EQUATION OF GENERAL POTENTIAL

A generalization of quantum mechanics is proposed, where the Lagrangian is the general form. The new quantum wave equation can describe the particle which is in general potential [Formula: see text], and the Schrödinger equation is only suited for the particle in common potential V(r, t). We think these new quantum wave equations can be used in some fields.