skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Foundations of Quantum Mechanics: Derivation of a dissipative Schrödinger equation from first principles

Abstract

Dissipation in Quantum Mechanics took some time to become a robust field of investigation after the birth of the field. The main issue hindering developments in the field is that the Quantization process was always tightly connected to the Hamiltonian formulation of Classical Mechanics. In this paper we present a quantization process that does not depend upon the Hamiltonian formulation of Classical Mechanics (although still departs from Classical Mechanics) and thus overcome the problem of finding, from first principles, a completely general Schrödinger equation encompassing dissipation. This generalized process of quantization is shown to be nothing but an extension of a more restricted version that is shown to produce the Schrödinger equation for Hamiltonian systems from first principles (even for Hamiltonian velocity dependent potential). - Highlights: • A Quantization process independent of the Hamiltonian formulation of quantum Mechanics is proposed. • This quantization method is applied to dissipative or absorptive systems. • A Dissipative Schrödinger equation is derived from first principles.

Authors:
;
Publication Date:
OSTI Identifier:
22617500
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 380; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CLASSICAL MECHANICS; HAMILTONIANS; QUANTIZATION; QUANTUM MECHANICS; SCHROEDINGER EQUATION

Citation Formats

Gonçalves, L.A., and Olavo, L.S.F., E-mail: olavolsf@gmail.com. Foundations of Quantum Mechanics: Derivation of a dissipative Schrödinger equation from first principles. United States: N. p., 2017. Web. doi:10.1016/J.AOP.2017.03.002.
Gonçalves, L.A., & Olavo, L.S.F., E-mail: olavolsf@gmail.com. Foundations of Quantum Mechanics: Derivation of a dissipative Schrödinger equation from first principles. United States. doi:10.1016/J.AOP.2017.03.002.
Gonçalves, L.A., and Olavo, L.S.F., E-mail: olavolsf@gmail.com. Mon . "Foundations of Quantum Mechanics: Derivation of a dissipative Schrödinger equation from first principles". United States. doi:10.1016/J.AOP.2017.03.002.
@article{osti_22617500,
title = {Foundations of Quantum Mechanics: Derivation of a dissipative Schrödinger equation from first principles},
author = {Gonçalves, L.A. and Olavo, L.S.F., E-mail: olavolsf@gmail.com},
abstractNote = {Dissipation in Quantum Mechanics took some time to become a robust field of investigation after the birth of the field. The main issue hindering developments in the field is that the Quantization process was always tightly connected to the Hamiltonian formulation of Classical Mechanics. In this paper we present a quantization process that does not depend upon the Hamiltonian formulation of Classical Mechanics (although still departs from Classical Mechanics) and thus overcome the problem of finding, from first principles, a completely general Schrödinger equation encompassing dissipation. This generalized process of quantization is shown to be nothing but an extension of a more restricted version that is shown to produce the Schrödinger equation for Hamiltonian systems from first principles (even for Hamiltonian velocity dependent potential). - Highlights: • A Quantization process independent of the Hamiltonian formulation of quantum Mechanics is proposed. • This quantization method is applied to dissipative or absorptive systems. • A Dissipative Schrödinger equation is derived from first principles.},
doi = {10.1016/J.AOP.2017.03.002},
journal = {Annals of Physics},
number = ,
volume = 380,
place = {United States},
year = {Mon May 15 00:00:00 EDT 2017},
month = {Mon May 15 00:00:00 EDT 2017}
}
  • A flotation model was developed by considering both hydrodynamic and surface forces involved in the process. The hydrodynamic forces were determined using a stream function and then used for estimating the kinetic energies that can be used for thinning the water films between bubbles and particles. The kinetic energies were compared with the energy barriers created by surface forces to determine the probability of adhesion. The surface forces considered included ion-electrostatic, London-van der Waals, and hydrophobic forces. Due to the insufficient information available on the hydrophobic forces for bubble-particle interactions, contributions from the hydrophobic force were back-calculated from the valuesmore » of the flotation rate constants determined experimentally with methylated silica sphered. The results show that the hydrophobic force constants (K{sub 132}) for bubble-particle interaction are larger than those (K{sub 131}) for particle-particle interactions but smaller than that (K{sub 232}) for air bubbles interacting with each other in the absence of surfactants. The K{sub 132} values determined in the present work are close to the geometric means of K{sub 131} and K{sub 232}, suggesting that the combining rules developed for dispersion forces may be useful for hydrophobic forces. The flotation rate equation derived in the present work suggests various methods of improving flotation processes.« less
  • The Planck aether hypothesis assumes that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying an exactly nonrelativistic law of motion. The Planck masses can be described by a quantum mechanical two-component nonrelativistic operator field equation having the form of a two-component nonlinear Schroedinger equation, with a spectrum of quasiparticles obeying Lorentz invariance as a dynamic symmetry for energies small compared to the Planck energy. We show that quantum mechanics itself can be derived from the Newtonian mechanics of the Planck aether as an approximate solution of Boltzmann`s equation for the locallymore » interacting positive and negative Planck masses, and that the validity of the nonrelativistic Schroedinger equation depends on Lorentz invariance as a dynamic symmetry. We also show how the many-body Schroedinger wave function can be factorized into a product of quasiparticles of the Planck aether with separable quantum potentials. Finally, we present a possible explanation of wave function collapse as a kind of enhanced gravitational collapse in the presence of the negative Planck masses.« less
  • We study exact solutions of the Schrödinger equation for some potentials. We introduce a parametric approach to supersymmetric quantum mechanics to calculate energy eigenvalues and corresponding wave functions exactly. As an application we solve Schrödinger equation for the generalized Morse potential, modified Hulthen potential, deformed Rosen-Morse potential and Poschl-Teller potential. The method is simple and effective to get the results.
  • The 2D space-fractional Schrödinger equation in the time-independent and time-dependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.
  • We study exact solutions of the Schrödinger equation for some potentials. We introduce a parametric approach to supersymmetric quantum mechanics to calculate energy eigenvalues and corresponding wave functions exactly. As an application we solve Schrödinger equation for the generalized Morse potential, modified Hulthen potential, deformed Rosen-Morse potential and Poschl-Teller potential. The method is simple and effective to get the results.