Mathematics of particle-wave mechanical systems / / James M. Hill

• Autore: Hill James M.
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (388 pages)
• Disciplina: 530.15
• Soggetto topico: Mathematical physics; Special relativity (Physics); Física matemàtica; Relativitat especial (Física)
• Soggetto genere / forma: Llibres electrònics
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Author's Foreword -- Contents -- 1 Introduction -- 1.1 Introduction -- 1.2 General Introduction -- 1.3 Special Relativity -- 1.4 Quantum Mechanics -- 1.5 de Broglie Particle-Wave Mechanics -- 1.6 Plan of Text -- 1.7 Tables of Major Symbols and Basic Equations -- 2 Special Relativity -- 2.1 Introduction -- 2.2 Lorentz Transformations -- 2.3 Einstein Addition of Velocities Law -- 2.4 Lorentz Invariances -- 2.5 Lorentz Invariant Velocity Fields u(x, t) -- 2.6 General Framework for Lorentz Invariances -- 2.7 Integral Invariants of the Lorentz Group -- 2.8 Alternative Validation of Lorentz Invariants -- 2.9 Jacobians of the Lorentz Transformations -- 2.10 Space-Time Transformation x'= ct and t' = x/c -- 2.11 The de Broglie Wave Velocity u'= c2/u -- 2.12 Force and Physical Energy Arising from Work Done -- 2.13 Lorentz Invariant Energy-Momentum Relations -- 2.14 Force Invariance for Constant Velocity Frames -- 2.15 Example: Motion in an Invariant Potential Field -- 2.16 Alternative Energy-Mass Velocity Variation -- 3 General Formulation and Basic Equations -- 3.1 Introduction -- 3.2 Louis Victor de Broglie -- 3.3 James Clerk Maxwell -- 3.4 Four Types of Matter and Variable Rest Mass -- 3.5 Modified Newton's Laws of Motion -- 3.6 Identity for Spatial Physical Force ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (bold f) /StPNE pdfmark [/StBMC pdfmarkfps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 3.7 Assumed Existence of Work Done Function ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (upper W left parenthesis bold x comma t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkW(x, t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark. ; 3.8 Forces ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (bold f) /StPNE pdfmark [/StBMC pdfmarkfps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and g Derivable from a Potential ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (upper V left parenthesis bold x comma t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkV(x, t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 3.9 Correspondence with Maxwell's Equations -- 3.10 Centrally or Spherically Symmetric Systems -- 3.11 Newtonian Kinetic Energy and Momentum -- 3.12 Newtonian Wave-Like Solution -- 3.13 Newtonian Work Done ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (upper W left parenthesis u comma lamda right parenthesis) /StPNE pdfmark [/StBMC pdfmarkW(u, λ)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark from ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (partial differential f divided by partial differential t equals c squared partial differential g divided by partial differential x) /StPNE pdfmark [/StBMC pdfmark∂f/∂t = c2 ∂g/∂xps: [/EMC pdfmark [/StPop pdf -- 4 Special Results for One Space Dimension -- 4.1 Introduction -- 4.2 Basic Equations -- 4.3 General Reformulations of Basic Equations -- 4.4 Important Identity -- 4.5 Formulation in Terms of Lorentz Invariants -- 4.6 Differential Relations for Invariants ξ and η -- 4.7 de Broglie's Guidance Equation -- 4.8 Vanishing of Force g in Direction of Time -- 4.9 Clairaut's Differential Equation with Parameter u -- 4.10 Hamiltonian for One Space Dimension -- 4.11 Lagrangian for One Space Dimension -- 5 Exact Wave-Like Solution -- 5.1 Introduction -- 5.2 Wave-Like Solution -- 5.3 Work Done W(u, λ) from ∂f/∂t = c2 ∂g/∂x -- 5.4 Simple Derivation of Wave-Like Solution -- 5.5 Relation to Solution of Special Relativity -- 5.6 Relation to Hubble Parameter. ; 5.7 Derivation of Integral for Hubble Formula -- 5.8 Dark Matter and Dark Energy as de Broglie States -- 6 Derivations and Formulae -- 6.1 Introduction -- 6.2 Derivation of Wave-Like Solution -- 6.3 Expressions for de Broglie Wave Energy -- 6.4 de Broglie Wave Energy for Particular λ -- 6.5 Alternative Approach to Evaluation of Integrals -- 6.6 Alternative Derivation for Wave Energy -- 6.7 Alternative Derivation of Exact Solution -- 6.8 Yet Another Approach to Evaluation of Integrals -- 7 Lorentz and Other Invariances -- 7.1 Introduction -- 7.2 Force Invariance Under Lorentz Transformations -- 7.3 Lorentz Invariance of ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (d script upper E divided by d p) /StPNE pdfmark [/StBMC pdfmarkd E/dpps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark or ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (d script upper E divided by d xi) /StPNE pdfmark [/StBMC pdfmarkdE/dξps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 7.4 Lorentz Invariance of Forces -- 7.5 Functional Dependence of Forces -- 7.6 Transformation x'= ct and t' = x/c -- 7.7 Force Invariance Under Superluminal Lorentz Frames -- 7.8 Particle and Wave Energies and Momenta -- 8 Further Results for One Space Dimension -- 8.1 Introduction -- 8.2 Wave Equation General Solution -- 8.3 Trivial Solution Only for Zero Spatial Force -- 8.4 Nontrivial Solutions for Zero Spatial Force -- 8.5 Generalisation of Wave-Like Solution -- 8.6 Solutions with Non-constant Rest Mass -- 8.7 Formulation for Variable Rest Mass. ; 8.8 Characteristics ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (alpha equals c t plus x) /StPNE pdfmark [/StBMC pdfmarkα= ct + xps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (beta equals c t minus x) /StPNE pdfmark [/StBMC pdfmarkβ= ct - xps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 8.9 p(x, t) and ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (script upper E left parenthesis x comma t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkE(x, t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Assumed Independent Variables -- 9 Centrally Symmetric Mechanical Systems -- 9.1 Introduction -- 9.2 Basic Equations with Spherical Symmetry -- 9.3 General Solutions for ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (script upper E left parenthesis r comma t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkE(r, t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and p(r, t) -- 9.4 Conservation of Energy ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (e plus script upper E plus upper V equals) /StPNE pdfmark [/StBMC pdfmarke + E + V = ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Constant -- 9.5 Fundamental Identity for f and g -- 9.6 ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (f equals plus or minus c left parenthesis g minus 2 p divided by r right parenthesis) /StPNE pdfmark [/StBMC pdfmarkf = c(g -2p/r)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Implies e0 Is Zero -- 9.7 Newtonian Gravitation and Schwarzschild Radius -- 9.8 Pseudo-Newtonian Gravitational Potential -- 9.9 Dark Matter-Dark Energy and Four Types of Matter. ; 9.10 Positive Energy (I) e = (e02 + (pc)2)1/2, ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (e 0 not equals 0) /StPNE pdfmark [/StBMC pdfmarke0 ≠0ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 9.11 Negative Energy (II) e = - (e02 + (pc)2)1/2, ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (e 0 not equals 0) /StPNE pdfmark [/StBMC pdfmarke0≠0ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 9.12 Positive Energy (III) e = pc, e0 = 0 -- 9.13 Negative Energy (IV) e = -pc, e0 = 0 -- 9.14 Similarity Stretching Solutions of Wave Equation -- 9.15 Some Examples Involving the Dirac Delta Function -- 9.16 Calculation Details for Similarity Solutions -- 9.17 de Broglie's Centrally Symmetric Guidance Formula -- 10 Relation with Quantum Mechanics -- 10.1 Introduction -- 10.2 Quantum Mechanics and Schrödinger Wave Equation -- 10.3 Group Velocity and de Broglie Waves -- 10.4 Lorentz Invariants ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (xi equals e x minus c squared p t) /StPNE pdfmark [/StBMC pdfmarkξ= ex - c2 ptps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (eta equals p x minus e t) /StPNE pdfmark [/StBMC pdfmarkη= px - etps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 10.5 Klein-Gordon Partial Differential Equation -- 10.6 Alternative Klein-Gordon-Schrödinger Equation -- 10.7 General Wave Structure of Solutions of Wave Equation -- 10.8 Wave Solutions of Klein-Gordon Equation -- 10.9 Time-Dependent Dirac Equation for Free Particle -- 11 Coordinate Transformations, Tensors and General Relativity -- 11.1 Summation Convention and Cartesian Tensors -- 11.2 Alternative Derivation of Basic Identity -- 11.3 General Curvilinear Coordinates -- 11.4 Partial Covariant Differentiation -- 11.5 Illustration for Single Space Dimension. ; 11.6 Formulae for Ricci and Einstein Tensors.
• Titolo autorizzato: Mathematics of Particle-Wave Mechanical Systems
• ISBN: 9783031197932; 9783031197925
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910633911203321
• Lo trovi qui: Univ. Federico II