scholarly journals A Variational Formulation for Relativistic Mechanics Based on Riemannian Geometry and its Application to the Quantum Mechanics Context

2021 ◽  
pp. 189-206
Author(s):  
Fabio Silva Botelho
Keyword(s):  
Quantum Mechanics ◽  
Riemannian Geometry ◽  
Variational Formulation

scholarly journals A note on Riemannian geometry and the relativistic quantum mechanics context

2018 ◽  
Vol 40 ◽  
pp. 58
Author(s):  
Fabio Silva Botelho
Keyword(s):  
Quantum Mechanics ◽  
Riemannian Geometry ◽  
Variational Formulation ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Basic Concepts

This article develops a variational formulation for  relativistic quantum mechanics. The main results are obtained through a connection between relativistic and quantum mechanics. Such a connection is established through basic concepts on Riemannian geometry and related extensions for the relativistic context.

scholarly journals A relativistic basis of the quantum theory.—II

1934 ◽  
Vol 145 (855) ◽  
pp. 645-656 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Quantum Theory ◽  
General Expression ◽  
Matrix Equation ◽  
Covariant Derivative ◽  
Riemannian Geometry ◽  
Order Equation ◽  
Second Order ◽  
Space Curvature

The paper is a continuation of the last paper communicated to these 'Proceedings.' In that paper, which we shall refer to as the first paper, a more general expression for space curvature was obtained than that which occurs in Riemannian geometry, by a modification of the Riemannian covariant derivative and by the use of a fifth co-ordinate. By means of a particular substitution (∆ μσ σ = 1/ψ ∂ψ/∂x μ ) it was shown that this curvature takes the form of the second order equation of quantum mechanics. It is not a matrix equation, however but one which has the character of the wave equation as it occurred in the earlier form of the quantum theory. But it contains additional terms, all of which can be readily accounted for in physics, expect on which suggested an identification with energy of the spin.

Wavelet Method in Numerical Modeling of Quantum Dots Embedded in Matrix

2013 ◽  
Vol 849 ◽  
pp. 427-434 ◽  
Author(s):  
Aleksander Muc ◽  
Aleksander Banaś
Keyword(s):  
Quantum Dots ◽  
Numerical Modeling ◽  
Quantum Mechanics ◽  
Analytical Method ◽  
Variational Formulation ◽  
Separation Of Variables ◽  
Rayleigh Quotient ◽  
Daubechies Wavelets ◽  
Wavelet Method

An analytical method of the solution of the governing nonlinear eigenproblem is proposed. It can be directly applied into the analysis of eigenstates in quantum mechanics. The method is based on the use of the separation of variables for specific shapes of quantum dots. In this way the analysis can be reduced to the disretization along one variable only the Daubechies wavelets. The eigenstates are derived with the use of the variational formulation combined with the method of the Rayleigh quotient

scholarly journals TOPOLOGICAL GAUGE THEORIES FROM SUPERSYMMETRIC QUANTUM MECHANICS ON SPACES OF CONNECTIONS

1993 ◽  
Vol 08 (03) ◽  
pp. 573-585 ◽  
Author(s):  
MATTHIAS BLAU ◽  
GEORGE THOMPSON
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Euler Characteristic ◽  
Moduli Spaces ◽  
Riemannian Geometry ◽  
Gauge Theories ◽  
Supersymmetric Quantum Mechanics ◽  
Three Dimensions ◽  
Flat Connections ◽  
Infinite Dimensional

We rederive the recently introduced N=2 topological gauge theories, representing the Euler characteristic of moduli spaces ℳ of connections, from supersymmetric quantum mechanics on the infinite-dimensional spaces [Formula: see text] of gauge orbits. To that end we discuss variants of ordinary supersymmetric quantum mechanics which have meaningful extensions to infinite-dimensional target spaces, and introduce supersymmetric quantum mechanics actions modeling the Riemannian geometry of submersions and embeddings, relevant to the projections [Formula: see text] and inclusions [Formula: see text] respectively. We explain the relation between Donaldson theory and the gauge theory of flat connections in three dimensions and illustrate the general construction by other two- and four-dimensional examples.

scholarly journals A Variational Formulation for the Relativistic Klein-Gordon Equation

2018 ◽  
Vol 40 ◽  
pp. 57
Author(s):  
Fabio Silva Botelho
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Variational Formulation ◽  
Gordon Equation ◽  
Function Concept ◽  
Normal Field ◽  
Klein Gordon ◽  
Definition Of ◽  
Classical And Quantum Mechanics ◽  
Klein Gordon Equation

This article develops a variational formulation for the relativistic Klein-Gordon equation.The main results are obtained through a connection between classical and quantum mechanics. Such a connection is established through the definition of  normal field and its relation with the wave function concept.

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