A NEW GEOMETRICAL FRAMEWORK FOR THE DE BROGLIE–BOHM QUANTUM THEORY

2013 ◽  
Vol 10 (03) ◽  
pp. 1250096 ◽  
Author(s):  
D. J. HURLEY ◽  
M. A. VANDYCK
Keyword(s):  
Quantum Theory ◽  
Bound States ◽  
Particle System ◽  
Zeeman Effect ◽  
Free Particle ◽  
Physical Space ◽  
Quantization Condition ◽  
Curvilinear Coordinates ◽  
Dimensional Manifold ◽  
De Broglie Bohm

A geometrical framework for the de Broglie–Bohm quantum theory is presented, in which the trajectories of an N-particle system are interpretable as the integral curves of a particular vector field defined on a 3N-dimensional manifold [Formula: see text] constructed from physical space M. It is mathematically valid even when M is curved. If M is flat, the usual theory is recovered and automatically expressed in whatever curvilinear coordinates one may wish to choose. The general construction is illustrated by the case of a free particle moving on the surface of a sphere. (A modified Bohr quantization condition for angular momentum is obtained, with a first correction proportional to the curvature.) The Zeeman effect and some bound states on the sphere are also considered.

scholarly journals A Quantum Space behind Simple Quantum Mechanics

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Chuan Sheng Chew ◽  
Otto C. W. Kong ◽  
Jason Payne
Keyword(s):  
Quantum Mechanics ◽  
Theoretical Model ◽  
Phase Space ◽  
Configuration Space ◽  
Center Of Mass ◽  
Free Particle ◽  
Physical Space ◽  
Dimensional Manifold ◽  
Quantum Space ◽  
Simple Quantum

In physics, experiments ultimately inform us about what constitutes a good theoretical model of any physical concept: physical space should be no exception. The best picture of physical space in Newtonian physics is given by the configuration space of a free particle (or the center of mass of a closed system of particles). This configuration space (as well as phase space) can be constructed as a representation space for the relativity symmetry. From the corresponding quantum symmetry, we illustrate the construction of a quantum configuration space, similar to that of quantum phase space, and recover the classical picture as an approximation through a contraction of the (relativity) symmetry and its representations. The quantum Hilbert space reduces into a sum of one-dimensional representations for the observable algebra, with the only admissible states given by coherent states and position eigenstates for the phase and configuration space pictures, respectively. This analysis, founded firmly on known physics, provides a quantum picture of physical space beyond that of a finite-dimensional manifold and provides a crucial first link for any theoretical model of quantum space-time at levels beyond simple quantum mechanics. It also suggests looking at quantum physics from a different perspective.

scholarly journals Three-particle system in a finite volume: formalism, quantization condition, spectrum and energy shift

2020 ◽  
Vol 241 ◽  
pp. 02005
Author(s):  
Jin-Yi Pang
Keyword(s):  
Finite Volume ◽  
Bound States ◽  
Particle System ◽  
Energy Shift ◽  
Effective Field ◽  
Quantization Condition ◽  
High Current ◽  
Scattering States ◽  
Three Body ◽  
Hadronic Process

Lattice QCD calculations provide an ab initio access to hadronic process. These calculations are usu ally performed in a small cubic volume with periodic boundary conditions. The infinite volume extrapolations for three-body systems are indispensable to understand many systems of high current interest. We derive the three-body quantization condition in a finite volume using an effective field theory in the particle-dimer picture. Our work shows a powerful and transparent method to read off three-body physical observables from lattice simulations. In this paper, we review the formalism, quantization condition, spectrum analysis and energy shifts calculation both for 3-body bound states and scattering states.

Constructing Quantum Mechanics

2019 ◽  
Author(s):  
Anthony Duncan ◽  
Michel Janssen
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Stark Effect ◽  
Zeeman Effect ◽  
Black Body ◽  
Black Body Radiation ◽  
Wave Mechanics ◽  
Specific Heats ◽  
Old Quantum Theory ◽  
Upper Level

This is the first of two volumes on the genesis of quantum mechanics. It covers the key developments in the period 1900–1923 that provided the scaffold on which the arch of modern quantum mechanics was built in the period 1923–1927 (covered in the second volume). After tracing the early contributions by Planck, Einstein, and Bohr to the theories of black‐body radiation, specific heats, and spectroscopy, all showing the need for drastic changes to the physics of their day, the book tackles the efforts by Sommerfeld and others to provide a new theory, now known as the old quantum theory. After some striking initial successes (explaining the fine structure of hydrogen, X‐ray spectra, and the Stark effect), the old quantum theory ran into serious difficulties (failing to provide consistent models for helium and the Zeeman effect) and eventually gave way to matrix and wave mechanics. Constructing Quantum Mechanics is based on the best and latest scholarship in the field, to which the authors have made significant contributions themselves. It breaks new ground, especially in its treatment of the work of Sommerfeld and his associates, but also offers new perspectives on classic papers by Planck, Einstein, and Bohr. Throughout the book, the authors provide detailed reconstructions (at the level of an upper‐level undergraduate physics course) of the cental arguments and derivations of the physicists involved. All in all, Constructing Quantum Mechanics promises to take the place of older books as the standard source on the genesis of quantum mechanics.

scholarly journals Solutions of the Relativistic Two-Body Problem. II. Quantum Mechanics

1972 ◽  
Vol 25 (2) ◽  
pp. 141 ◽  
Author(s):  
JL Cook
Keyword(s):  
Bound States ◽  
Free Particle ◽  
Plane Waves ◽  
Addition Theorem ◽  
Partial Wave Analysis ◽  
Harmonic Oscillator Potential ◽  
Two Body Problem ◽  
Probability Densities ◽  
Square Well ◽  
Potential Harmonic

This paper discusses the formulation of a quantum mechanical equivalent of the relative time classical theory proposed in Part I. The relativistic wavefunction is derived and a covariant addition theorem is put forward which allows a covariant scattering theory to be established. The free particle eigenfunctions that are given are found not to be plane waves. A covariant partial wave analysis is also given. A means is described of converting wavefunctions that yield probability densities in 4-space to ones that yield the 3-space equivalents. Bound states are considered and covariant analogues of the Coulomb potential, harmonic oscillator potential, inverse cube law of force, square well potential, and two-body fermion interactions are discussed.

Integrable and chaotic motions of four vortices II. Collision dynamics of vortex pairs

1988 ◽  
Vol 326 (1593) ◽  
pp. 655-696 ◽  
Keyword(s):  
Bound States ◽  
Degrees Of Freedom ◽  
Complex Structure ◽  
Physical Space ◽  
Vantage Point ◽  
Collision Dynamics ◽  
Chaotic Motions ◽  
Vortex Pairs ◽  
Scattering States ◽  
Space Mechanisms

The interaction of two vortex pairs is investigated analytically and by numerical experiments from the vantage point of dynamical-systems theory. Vortex pairs can escape to infinity, so the phase space of this system is unbounded in contrast to that of four identical vortices investigated previously. Chaotic motion is nevertheless possible both for ‘bound states’ of the system and for ‘scattering states’. For the bound states standard Poincare section techniques suffice. For scattering states chaos appears as complex structure in the numerically generated plot of scattering angle against impact parameter. Interpretations of physical space mechanisms leading to chaos are given. Analytical characterizations of the system include a formal reduction to two degrees of freedom by canonical transformations and an identification and discussion of integrable cases of which one is apparently new.

GROUP THEORY CLASSIFICATION SCHEMES OF THE N-ELECTRON QUANTUM DOT

1994 ◽  
Vol 08 (09) ◽  
pp. 1159-1189
Author(s):  
R.W. HAASE ◽  
N.F. JOHNSON
Keyword(s):  
Bound States ◽  
Particle System ◽  
Discrete Series ◽  
Series Representation ◽  
Dynamical Symmetry ◽  
Pauli Principle ◽  
General Terms ◽  
Discrete Series Representation ◽  
Electron Quantum ◽  
The Many

We develop a general framework for discussing collective behavior in confined many-electron systems. Our specific goal is the application to N-electron quantum dots, which are mesoscopic semiconductor systems of great current interest as possible ultra-small electronic devices. In view of its broad applicability, we are able to cast the discussion of the many-electron problem in general terms. We consider the general N-interacting particle system in d dimensions and study its bilinear dynamical symmetry group which is the noncompact symplectic group Sp(2Nd, R). Giving their explicit dependence on N and d, we focus on the classification of many-particle bound states which requires knowledge of the unitary discrete series representation theory of Sp(2Nd, R) and the corresponding character reductions. We also discuss matrix elements of the generators, the implementation of the Pauli principle, and a procedure to derive total angular momentum quantum numbers associated with a given total spin.

TOPOLOGICAL SOLUTION OF BOHMIAN QUANTUM MECHANICS

2009 ◽  
Vol 24 (06) ◽  
pp. 453-461 ◽  
Author(s):  
XUGUANG SHI ◽  
MING YU ◽  
YISHI DUAN
Keyword(s):  
Quantum Mechanics ◽  
Particle System ◽  
World Line ◽  
Zero Point ◽  
Current Theory ◽  
Brouwer Degree ◽  
Topological Current ◽  
The World ◽  
Bohmian Quantum Mechanics ◽  
De Broglie Bohm

The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrödinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer degree. The evolution of the world line at the bifurcation point is given.

ON THE GEOMETRIZATION OF BOHMIAN MECHANICS: A NEW APPROACH TO QUANTUM GRAVITY

1998 ◽  
Vol 13 (04) ◽  
pp. 677-693 ◽  
Author(s):  
FATIMAH SHOJAI ◽  
MEHDI GOLSHANI
Keyword(s):  
Quantum Theory ◽  
Quantum Gravity ◽  
Classical Limit ◽  
Bohmian Mechanics ◽  
Present Theory ◽  
Space Time ◽  
New Approach ◽  
Theory Of Motion ◽  
Quantum Theory Of Motion ◽  
De Broglie Bohm

In this paper, a new approach to quantum gravity is presented in which the de-Broglie–Bohm quantum theory of motion is geometrized. This way of considering quantum gravity leads automatically to the fact that the quantum effects are contained in the conformal degree of freedom of the space–time metric. The present theory is then applied to the maximally symmetric space–time of cosmology, and it is observed that it is possible to avoid the initial singularity, while at large times the correct classical limit emerges.

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