Advances in Imaging and Electron Physics (eBook)

Front Cover 1
Dirac's Difference Equation and the Physics of Finite Differences 4
Copyright Page 5
Contents 8
Preface 10
Future contributions 12
Foreword 16
List of frequently used symbols 18
Chapter 1. Introduction 20
1.1 Maxwell’s Equations with Magnetic Dipole Currents 20
1.2 Lagrange Function 28
1.3 Basic Concepts of the Calculus of Finite Differences 34
1.4 Finite Differences and Spatial Dimensions 41
1.5 Curved Space in a Difference Theory 56
Chapter 2. Modified Dirac Equation 66
2.1 Differential Equation with Magnetic Current Density 66
2.2 Modified Dirac Difference Equation 77
2.3 Solution of the Difference Equation for .x0 
87 
2.4 Time Variation of .01 (., .) 98
2.5 Hamiltonian Formalism and Quantization 110
2.6 Finite Limit of the Period Number . 
118 
Chapter 3. Inhomogeneous Dirac Difference Equation 127
3.1 Inhomogeneous Equation (2.2-33) 127
3.2 Resolution .x< <
3.3 Quantization of the Solution 151
3.4 Evaluation of the Energy U for Small Distances .x 159
Chapter 4. Dirac Difference Equation in Spherical Coordinates 177
4.1 Electron in an Electromagnetic Field 177
4.2 Relativistic Lagrange Function 182
4.3 Quantization of the Homogeneous Equation 190
4.4 Unbounded Electron in a Coulomb Field 198
4.5 Anti-Particles With s0 Near s = –1 203
4.6 Solutions for s0 Near s = i 209
4.7 Solutions with s0 in the Neighborhood of s = – i 220
4.8 Energy or Mass Ratios for E/m0c2 > 1 and .r min
Chapter 5. Inhomogeneous Equations for Coulomb Potential 241
5.1 Quantization of the Inhomogeneous Term 241
5.2 Separation of the Functions .1j(.,.) 
251 
5.3 Solutions for v(.) and w(.) 
254 
Chapter 6. Appendix 260
6.1 Calculations for Section 2.3 260
6.2 Calculations for Section 3.2 262
6.3 Inhomogeneous Difference Equations 266
6.4 Further Elaboration of Eq.(2.1-28) 277
6.5 Quantization of Lcr2 to Lcf5 279
6.6 Polynomials as Solutions of Difference Equations of Second Order 281
6.7 Separation of Variables in Section 4.3 290
6.8 Solution of .1(.) and .4(.) 
298 
6.9 Calculation of C1v and C4v in Section 6.7 312
6.10 Slanted Coordinate Systems With Finite Differences 318
6.11 Riemann Manifolds and Bended Eigen-Coordinates 320
References and Bibliography 329
Index 335