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 |  موضوع: كتاب Mathematics of Particle - Wave Mechanical Systems السبت 04 فبراير 2023, 2:43 am | |
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Mathematics of Particle - Wave Mechanical Systems
James M. Hill
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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
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