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أخواني في الله أحضرت لكم كتاب Mathematics of Particle - Wave Mechanical Systems James M. Hill
![كتاب Mathematics of Particle - Wave Mechanical Systems M_o_p_11](https://i.servimg.com/u/f61/15/36/74/14/m_o_p_11.jpg) و المحتوى كما يلي :
Contents 1 Introduction . 1 1.1 Introduction . 1 1.2 General Introduction . 1 1.3 Special Relativity . 4 1.4 Quantum Mechanics . 5 1.5 de Broglie Particle-Wave Mechanics 7 1.6 Plan of Text . 9 1.7 Tables of Major Symbols and Basic Equations . 12 2 Special Relativity . 15 2.1 Introduction . 15 2.2 Lorentz Transformations . 16 2.3 Einstein Addition of Velocities Law . 18 2.4 Lorentz Invariances 20 2.5 Lorentz Invariant Velocity Fields u(x, t) 24 2.6 General Framework for Lorentz Invariances 25 2.7 Integral Invariants of the Lorentz Group 28 2.8 Alternative Validation of Lorentz Invariants 31 2.9 Jacobians of the Lorentz Transformations 32 2.10 Space-Time Transformation x = ct and t = x/c . 34 2.11 The de Broglie Wave Velocity u = c2/u . 35 2.12 Force and Physical Energy Arising from Work Done 37 2.13 Lorentz Invariant Energy-Momentum Relations 40 2.14 Force Invariance for Constant Velocity Frames . 44 2.15 Example: Motion in an Invariant Potential Field . 45 2.16 Alternative Energy-Mass Velocity Variation 49 3 General Formulation and Basic Equations 55 3.1 Introduction . 55 3.2 Louis Victor de Broglie 56 3.3 James Clerk Maxwell 63 3.4 Four Types of Matter and Variable Rest Mass 68 xixii Contents 3.5 Modified Newton’s Laws of Motion 71 3.6 Identity for Spatial Physical Force f . 73 3.7 Assumed Existence of Work Done Function W(x, t) 75 3.8 Forces f and g Derivable from a Potential V (x, t) 79 3.9 Correspondence with Maxwell’s Equations . 80 3.10 Centrally or Spherically Symmetric Systems . 83 3.11 Newtonian Kinetic Energy and Momentum 85 3.12 Newtonian Wave-Like Solution . 86 3.13 Newtonian Work Done W(u, λ) from ∂f /∂t = c2∂g/∂x 88 4 Special Results for One Space Dimension . 91 4.1 Introduction . 91 4.2 Basic Equations 92 4.3 General Reformulations of Basic Equations 96 4.4 Important Identity 100 4.5 Formulation in Terms of Lorentz Invariants 102 4.6 Differential Relations for Invariants ξ and η 113 4.7 de Broglie’s Guidance Equation . 118 4.8 Vanishing of Force g in Direction of Time 127 4.9 Clairaut’s Differential Equation with Parameter u 135 4.10 Hamiltonian for One Space Dimension . 145 4.11 Lagrangian for One Space Dimension 147 5 Exact Wave-Like Solution . 149 5.1 Introduction . 149 5.2 Wave-Like Solution 150 5.3 Work Done W(u, λ) from ∂f /∂t = c2∂g/∂x . 152 5.4 Simple Derivation of Wave-Like Solution 154 5.5 Relation to Solution of Special Relativity . 155 5.6 Relation to Hubble Parameter . 155 5.7 Derivation of Integral for Hubble Formula 156 5.8 Dark Matter and Dark Energy as de Broglie States . 158 6 Derivations and Formulae . 163 6.1 Introduction . 163 6.2 Derivation of Wave-Like Solution . 165 6.3 Expressions for de Broglie Wave Energy . 167 6.4 de Broglie Wave Energy for Particular λ 171 6.5 Alternative Approach to Evaluation of Integrals 173 6.6 Alternative Derivation for Wave Energy 175 6.7 Alternative Derivation of Exact Solution 178 6.8 Yet Another Approach to Evaluation of Integrals 180 7 Lorentz and Other Invariances . 183 7.1 Introduction . 183 7.2 Force Invariance Under Lorentz Transformations 185 7.3 Lorentz Invariance of dE /dp or dE /dξ 187Contents xiii 7.4 Lorentz Invariance of Forces . 189 7.5 Functional Dependence of Forces . 191 7.6 Transformation x = ct and t = x/c . 194 7.7 Force Invariance Under Superluminal Lorentz Frames . 197 7.8 Particle and Wave Energies and Momenta 199 8 Further Results for One Space Dimension . 205 8.1 Introduction . 205 8.2 Wave Equation General Solution 205 8.3 Trivial Solution Only for Zero Spatial Force . 207 8.4 Nontrivial Solutions for Zero Spatial Force . 208 8.5 Generalisation of Wave-Like Solution 211 8.6 Solutions with Non-constant Rest Mass 214 8.7 Formulation for Variable Rest Mass . 219 8.8 Characteristics α = ct + x and β = ct − x 224 8.9 p(x, t) and E (x, t) Assumed Independent Variables . 230 9 Centrally Symmetric Mechanical Systems . 233 9.1 Introduction . 233 9.2 Basic Equations with Spherical Symmetry . 234 9.3 General Solutions for E (r, t) and p(r, t) 236 9.4 Conservation of Energy e + E + V = Constant 238 9.5 Fundamental Identity for f and g . 240 9.6 f = ±c(g − 2p/r) Implies e0 Is Zero 244 9.7 Newtonian Gravitation and Schwarzschild Radius . 246 9.8 Pseudo-Newtonian Gravitational Potential 247 9.9 Dark Matter-Dark Energy and Four Types of Matter . 250 9.10 Positive Energy (I) e = (e02 + (pc)2)1/2, e0 = 0 251 9.11 Negative Energy (II) e = −(e02 + (pc)2)1/2, e0 = 0 253 9.12 Positive Energy (III) e = pc, e0 = 0 254 9.13 Negative Energy (IV) e = −pc, e0 = 0 . 257 9.14 Similarity Stretching Solutions of Wave Equation 259 9.15 Some Examples Involving the Dirac Delta Function . 261 9.16 Calculation Details for Similarity Solutions 269 9.17 de Broglie’s Centrally Symmetric Guidance Formula 276 10 Relation with Quantum Mechanics . 281 10.1 Introduction . 281 10.2 Quantum Mechanics and Schrödinger Wave Equation . 282 10.3 Group Velocity and de Broglie Waves 285 10.4 Lorentz Invariants ξ = ex − c2pt and η = px − et 286 10.5 Klein–Gordon Partial Differential Equation 289 10.6 Alternative Klein–Gordon–Schrödinger Equation 291 10.7 General Wave Structure of Solutions of Wave Equation . 294 10.8 Wave Solutions of Klein–Gordon Equation . 298 10.9 Time-Dependent Dirac Equation for Free Particle . 302xiv Contents 11 Coordinate Transformations, Tensors and General Relativity 305 11.1 Summation Convention and Cartesian Tensors . 306 11.2 Alternative Derivation of Basic Identity 309 11.3 General Curvilinear Coordinates 310 11.4 Partial Covariant Differentiation . 316 11.5 Illustration for Single Space Dimension 324 11.6 Formulae for Ricci and Einstein Tensors 327 11.7 Two Illustrative Line Elements 340 11.8 Spiral Gravitating Structures . 345 12 Conclusions, Summary and Postscript . 361 12.1 Introduction . 361 12.2 Conclusions . 362 12.3 Summary 366 12.4 Postscript . 368 Bibliography 371 Index . 375 Index A Alternative derivation for wave energy, 175–178 Alternative derivation of basic identity, 309 Alternative derivation of exact solution, 178–180 Alternative energy-mass velocity variation, 49–54 Alternative evaluation of integrals, 173–175 Alternative Klein–Gordon–Schrödinger equation, 291–294 Another approach to evaluation of integrals, 180–182 B Bacon, F. Sir, 369 Basic equations, 9–14, 55–89, 91–100, 165, 179, 183, 184, 189, 195, 197, 198, 205, 214, 219, 225, 233, 355, 367 spherical symmetry, 234–236 Bianchi identity, 305, 322, 323 Bohm, D., 16, 61, 369 Bush, J.W.M., 8, 362 C Centrally symmetric, 11, 13, 14, 234, 276–280, 316 mechanical systems, 72, 78, 94, 99, 122, 233–280, 363 Characteristics, 3–5, 8, 11, 13, 14, 17, 21, 23, 24, 28–30, 35, 58, 93, 102, 109, 111, 113, 120–122, 125, 128, 130, 131, 134, 149, 161, 189, 193, 205, 208, 211, 224–232, 236, 237, 240, 242, 255, 266, 277, 282, 289, 291, 292, 294, 295, 297, 299, 346, 347, 367 Christoffel symbols, 12, 305, 310, 316, 318, 319, 345, 357 first kind, 319, 324, 329, 330, 341, 343, 348, 353 second kind, 318, 319, 325, 330, 335, 342, 345, 353 Clairaut’s differential equation, 10, 92, 122, 127, 135–142 Cohen, B., 5, 370 Conservation of energy, 10, 14, 55, 79, 80, 102, 111–113, 154, 161, 195, 228, 238–240, 243, 246–248, 281, 283, 364 Coordinate transformations, 11, 35, 42, 305–360 Correspondence with Maxwell’s equations, 10, 80–82 Couder, Y., 8, 362 Curvature invariant, 322, 327, 336–338, 341, 342, 344, 351 Curvilinear coordinates, 12, 305, 306, 310, 311 D Dark energy, 1, 3–5, 9, 69, 89, 91, 101, 102, 158–162, 224, 233, 243, 250–251, 257, 286, 365, 367 Dark energy as de Broglie state, 149, 158–162 Dark matter, 1–5, 9, 69, 89, 91, 101, 149, 158–162, 224, 233, 243, 250–251, 254, 365, 367 184, 191–194 Fundamental force identity, 233, 240–244 G General curvilinear coordinates, 12, 305, 310–317 Generalisation of wave-like solution, 11, 205, 211–214 General relativity, 2, 8, 9, 11, 12, 15, 52, 112, 252, 305–360, 364 General solution for energy, 150 General solution for momentum, 111, 150 General wave structure, 294–298 Goldstein, S., 369 Gravitational potential, 233, 247–249, 256 Group velocity, 3, 36, 58, 285–287 H Hamiltonian, 10, 46, 79, 92, 145–147 Heaviside unit step function, 70, 214 Hubble parameter, 2, 10, 149, 155–157, 367 I Identity, 10, 18, 55, 91, 150, 178, 188, 233, 282, 305, 363 basic, 305, 309, 313 Christoffel symbol, 357–360 fundamental, 233, 240–244, 308, 309 important, 55, 100–102, 108 spatial force, 14 Integral for Hubble formula, 156–157 Invariant potential field, 15, 45–49 K Klein-Gordon partial differential equation, 11, 282, 289–291, 294 Kronecker delta, 307–311, 318 Kuhn, T.S., 369 L Lagrangian, 10, 65, 79, 92, 145, 147 Laplacian, 84, 315, 321, 337 Levi-Civita symbol, 307, 308 Line element, 113, 114, 306, 310, 324, 327, 329, 340–347 Lorentz group, integral invariants, 28–31 Lorentz invariance, 2, 5, 9, 13, 15, 17, 18, 20–23, 41, 51, 68, 72, 149, 151, 163,Index 377 167, 183–204, 217, 219, 222, 282, 283, 362, 363, 365, 366 forces, 75, 149, 163, 183, 184, 186, 189–191, 365 general framework, 25–28 Lorentz invariant energy-momentum, 5, 15, 38, 40–44, 73, 183–185, 198, 283 Lorentz invariant velocity, 15, 24–25 Lorentz transformation, 5, 9, 10, 15–20, 22–24, 26, 38, 42, 45, 72, 116, 118, 142, 183–191, 193, 197, 198, 200, 289, 363, 366, 368, 370 Jacobian, 32–34 M Maxwell, J.C., 8, 10, 55, 63–68, 368 Maxwell’s equations, 5, 8, 61–63, 65, 66, 80–82, 363 Modified Newton’s laws of motion, 71–73, 363 Murray Gell-Mann, 5 N Negative energy, 151, 253–254, 257–259, 365 Newtonian gravitation, 246–247 Newtonian kinetic energy, 85–86 Newtonian momentum, 85–86 Newtonian wave-like solution, 86–88 Newtonian work done, 88–89 Nontrivial solutions for zero spatial force, 205, 208–211 P Partial covariant differentiation, 12, 305, 312, 316–323 Particle and wave energies and momenta, 199–204 Planck data, 1, 158, 159 Planck’s constant, 6, 8, 58, 59, 72, 284 Positive energy, 251–257 Pseudo-Newtonian gravitational potential, 233, 247–249 Q Quantum mechanics, 2, 5–7, 9, 11, 59, 63, 91, 145, 204, 281–304, 362, 368–370 operator relations, 6, 11, 281–284, 368 R Reformulation of equations, 10, 91, 96–100 Relation with quantum mechanics, 281–304 S Schrödinger wave equation, 2, 6, 9, 11, 281–284, 292, 368, 369 Schwarzschild radius, 11, 233, 246–247 Similarity solutions, 91, 93, 95, 187, 234, 245–246, 256–257, 259–261, 263, 269–276, 345 Similarity stretching solutions of wave equation, 259–261 Solutions with non-constant rest mass, 109, 214–219 Space-time transformation, 34–35, 86, 103, 118, 142, 184, 194, 197, 200, 203, 370 Special relativity, 1, 2, 4–5, 7, 9–11, 15–54, 58, 63, 66, 68, 69, 71, 72, 78, 79, 149, 155, 183, 198, 200, 219, 281, 286, 289, 362, 363, 366, 368–370 Spherical symmetry, 234–236 Spiral gravitating structure, 323, 345–360 Summation convention, 12, 306–308, 310, 334 Synge, J.L., 8 T Tensor, 11, 12, 305–360 Cartesian, 12, 74, 305–308, 310 covariant curvature, 12, 305, 306, 322, 325–327, 335–337, 339, 341, 343, 348, 351 Einstein, 12, 305, 306, 322, 323, 327–340, 342, 344, 346, 351, 352, 354, 357 metric, 306, 310–313, 315, 318–321, 323–329, 334, 335, 341, 343, 345–347, 351, 352, 354, 364 Ricci, 12, 305, 306, 322, 326–340 Riemann-Christoffel, 322, 327 Trivial solution only for zero spatial force, 207–208 V Vanishing of force in direction of time, 127–135 Variable rest mass, 11, 68–71, 214, 219–224 Vector product, 307, 308, 314 Voigt, W., 370 W Walking-drop system, 8, 362 Wave equation, 2, 16, 64, 91, 153, 170, 184, 205, 234, 281, 315, 363378 Index Wave equation (cont.) general solution, 93, 153, 205–207, 227, 294, 297 Schrodinger, 2, 6, 9, 11, 281–284, 292, 368, 369 Wave solutions of Klein-Gordon equation, 298–302 Weinberger, P., 8, 369 Work done function, 55, 56, 74–80 #ماتلاب,#متلاب,#Matlab,
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