Solitons and Quantum Behavior Abstract
Total Page:16
File Type:pdf, Size:1020Kb
Solitons and Quantum Behavior Richard A. Pakula email: [email protected] April 9, 2017 Abstract In applied physics and in engineering intuitive understanding is a boost for creativity and almost a necessity for efficient product improvement. The existence of soliton solutions to the quantum equations in the presence of self-interactions allows us to draw an intuitive picture of quantum mechanics. The purpose of this work is to compile a collection of models, some of which are simple, some are obvious, some have already appeared in papers and even in textbooks, and some are new. However this is the first time they are presented (to our knowledge) in a common place attempting to provide an intuitive physical description for one of the basic particles in nature: the electron. The soliton model applied to the electrons can be extended to the electromagnetic field providing an unambiguous description for the photon. In so doing a clear image of quantum mechanics emerges, including quantum optics, QED and field quantization. To our belief this description is of highly pedagogical value. We start with a traditional description of the old quantum mechanics, first quantization and second quantization, showing the tendency to renounce to 'visualizability', as proposed by Heisenberg for the quantum theory. We describe an intuitive description of first and second quantization based on the concept of solitons, pioneered by the 'double solution' of de Broglie in 1927. Many aspects of first and second quantization are clarified and visualized intuitively, including the possible achievement of ergodicity by the so called ‘vacuum fluctuations’. PACS: 01.70.+w, +w, 03.65.Ge , 03.65.Pm, 03.65.Ta, 11.15.-q, 12.20.*, 14.60.Cd, 14.70.-e, 14.70.Bh, 14.70.Fm, 14.70.Hp, 14.80.Bn, 32.80.y, 42.50.Lc 1 Table of Contents Solitons and Quantum Behavior ............................................................................................................................................. 1 Abstract ................................................................................................................................................................................... 1 1. Chapter: Quantum Mechanics ........................................................................................................................................ 7 1.1. Classical physics .......................................................................................................................................................... 7 1.2. Quantum mechanics. .................................................................................................................................................. 7 1.2.1. Old Quantum Theory .......................................................................................................................................... 7 1.2.2. First quantization ................................................................................................................................................ 8 1.2.3. Second quantization.......................................................................................................................................... 11 1.3. Intuitive interpretation ............................................................................................................................................. 13 1.3.1. First quantization .............................................................................................................................................. 13 1.3.2. Second quantization.......................................................................................................................................... 15 1.4. Conclusion ................................................................................................................................................................. 18 2. Chapter: Dirac-Poisson Soliton ..................................................................................................................................... 20 2.1. The Quantum Equations. Traditional Presentation. ................................................................................................ 20 2.1.1. Schrödinger equation ........................................................................................................................................ 20 2.1.1.1. Equation .................................................................................................................................................... 20 2.1.1.2. Hamiltonian ............................................................................................................................................... 20 2.1.1.3. Charge density........................................................................................................................................... 20 2.1.2. Pauli equation: .................................................................................................................................................. 21 2.1.2.1. Spinors ....................................................................................................................................................... 21 2.1.2.2. Equation .................................................................................................................................................... 22 2.1.2.3. Charge density........................................................................................................................................... 23 2.1.2.4. Alternative interpretation, Dipole approximation .................................................................................... 23 2.1.3. Klein Gordon equation ...................................................................................................................................... 24 2.1.3.1. Equation .................................................................................................................................................... 24 2.1.3.2. Hamiltonian ............................................................................................................................................... 24 2.1.3.3. Charge and current density: ...................................................................................................................... 25 2.1.4. Klein Gordon equation in Schrödinger form ..................................................................................................... 25 2.1.4.1. Linearization of energy: ............................................................................................................................ 25 2.1.4.2. Linear Equation ......................................................................................................................................... 27 2.1.4.3. The Feshbach-Villars representation ........................................................................................................ 29 2.1.5. Dirac Equation. .................................................................................................................................................. 29 2.1.5.1. Traditional ................................................................................................................................................. 29 2 2.1.5.2. Decoding the traditional Dirac equation ................................................................................................... 31 2.1.6. Maxwell equations ............................................................................................................................................ 37 2.1.6.1. Equations................................................................................................................................................... 37 2.1.6.2. Hamiltonian ............................................................................................................................................... 37 2.2. The Chain Model. ...................................................................................................................................................... 37 2.2.1. Klein Gordon Equation ...................................................................................................................................... 37 2.3. De Broglie’s Guiding Condition and Bohm’s Quantum Potential ............................................................................. 43 2.3.1. Introduction ...................................................................................................................................................... 43 2.3.1.1. Effects of Boundary conditions in the absence of external fields, De Broglie .......................................... 43 2.3.1.2. External fields, Bohm ................................................................................................................................ 47 2.3.2. Schrödinger Equation ........................................................................................................................................ 50 2.3.3. Klein Gordon equation ...................................................................................................................................... 52 2.3.4. Pauli Equation ................................................................................................................................................... 54 2.3.4.1. Pauli Quantum Potential ..........................................................................................................................