The Dirac Equation and its Solutions (eBook)

Vladislav G. Bagrov, Dmitry Gitman (Autoren)

eBook Download: PDF | EPUB

2014
441 Seiten
De Gruyter (Verlag)
978-3-11-037775-0 (ISBN)

Infos

The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics, representing the one-particle wave equation of motion for electrons in an external electromagnetic field.
In this monograph, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.

Vladislav Bagrov, Tomsk State University, Russia;

Dmitry Gitman, University of Sao Paulo, Brazil.

Autor

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Vladislav Bagrov, Tomsk State University, Russia;

Dmitry Gitman, University of Sao Paulo, Brazil.

Inhalt

Contents 7
Preface 5
Acknowledgements 6
1 Introduction 13
1.1 Book content 14
1.2 Notation 16
2 Relativistic equations of motion 21
2.1 Classical equations 21
2.1.1 Maxwell equations 21
2.1.2 Equations of motion for a charge in an electromagnetic field 23
2.1.3 Hamilton–Jacobi equation 24
2.2 K–G equation 24
2.2.1 General 24
2.2.2 Evolution function and completeness relations 27
2.2.3 Hamiltonian forms of the K–G equation 28
2.3 Dirac equation 29
2.3.1 General 29
2.3.2 Evolution function and completeness relation 33
2.3.3 Reducing Dirac equation into two independent sets of second-order equations for spinors 35
2.3.4 Reducing Dirac equation into two independent sets of fourth-order equations for scalar functions 36
2.3.5 Squaring the Dirac equation 37
2.4 Spin operators 41
3 Basic exact solutions 49
3.1 Free particle motion 49
3.1.1 Classical motion 49
3.1.2 States with a given momentum 50
3.1.3 Positive and negative frequency solutions 53
3.1.4 Light-cone variables and coherent states 54
3.1.5 States with given angular momentum projection 60
3.2 Particles in plane-wave field 64
3.2.1 Plane-wave electromagnetic field 64
3.2.2 Classical motion in the plane-wave field 65
3.2.3 Quantum motion in plane-wave field 67
3.3 Particles in BGY field 72
3.3.1 BGY field 72
3.3.2 Classical motion in a BGY field 72
3.3.3 Quantum motion in a BGY field 75
3.4 Particles in a constant and uniform magnetic field 76
3.4.1 Introduction 76
3.4.2 Page´s and Rabi´s solutions 78
3.4.3 Creation and annihilation operators 82
3.4.4 Stationary states 85
3.4.5 Orthonormality and completeness of stationary states 94
3.4.6 Coherent states 98
3.4.7 Zero magnetic field limit 103
3.4.8 Some other types of nonstationary states 104
3.5 Particles in spherically symmetric fields 106
3.5.1 General 106
3.5.2 Separation of variables in K–G and Dirac equations 109
3.5.3 Specification of potentials and complete classical solution 111
3.5.4 Azimuthal motion 114
3.5.5 Radial motion 119
3.6 Particles in the Aharonov–Bohm field and in its superpositions with other fields 125
3.6.1 Introduction 125
3.6.2 Aharonov–Bohm field 128
3.6.3 Magnetic-solenoid field 130
3.6.4 Quasicoherent states in the magnetic-solenoid field 139
3.6.5 Aharonov–Bohm field and additional electromagnetic fields 146
4 Particles in fields of special structure 156
4.1 Introduction 156
4.2 Crossed electromagnetic fields 157
4.2.1 General 157
4.2.2 Stationary crossed fields 160
4.2.3 Nonstationary crossed fields 164
4.3 Longitudinal electromagnetic fields 178
4.3.1 General 178
4.3.2 Longitudinal motion in the electric field 181
4.3.3 Transversal motion in the magnetic field 184
4.4 Superposition of crossed and longitudinal fields 186
4.4.1 General 186
4.4.2 Crossed and longitudinal electric field 187
4.4.3 Crossed and longitudinal electric and magnetic fields 191
4.5 Fields of nonstandard structure 214
5 Dirac–Pauli equation and its solutions 226
5.1 Introduction 226
5.2 Constant and uniform magnetic field 227
5.3 Plane-wave field 229
5.4 Superposition of a plane-wave field and a parallel electric field 232
6 Propagators of relativistic particles 235
6.1 Introduction 235
6.2 Proper-time representations for particle propagators 237
6.2.1 General 237
6.2.2 Proper-time representations in a constant uniform field and a plane wave field 240
6.3 Path-integrals for particle propagators 245
6.3.1 Path integral for K–G propagator 246
6.3.2 Path integral for the Dirac propagator in even dimensions 249
6.3.3 Path integral for the Dirac propagator in odd dimensions 253
6.3.4 Classical and pseudoclassical description of relativistic particles 255
6.4 Calculations of Dirac propagators using path integrals 256
6.4.1 Spin factor in 3 + 1 dimensions 256
6.4.2 Propagator in the constant uniform electromagnetic field 260
6.4.3 Propagator in a constant uniform field and a plane wave field 264
6.4.4 Propagator in a constant uniform field in 2 + 1 dimensions 270
7 Electron interacting with a quantized electromagnetic plane wave 275
7.1 Dirac equation with quantized plane wave 275
7.1.1 General 275
7.1.2 Separation of variables 279
7.2 Quantized monochromatic plane wave with arbitrary polarization 283
7.3 Quantized plane wave of general form 285
7.4 Canonical forms for Hamiltonian of quasiphotons 288
7.5 Stationary and coherent states 297
7.5.1 Stationary states 297
7.5.2 Relations of orthogonality, normalization and completeness 300
7.6 Reduction to Volkov solutions 302
7.7 Electron interacting with quantized plane-wave and with external electromagnetic background 304
7.7.1 Classical plane wave along the quantized field 304
7.7.2 Classical magnetic field directed along the quantized plane wave 308
7.8 Linear and quadratic combinations of creation and annihilation operators 311
7.8.1 Linear combinations 311
7.8.2 Quadratic combinations 318
8 Spin equation and its solutions 322
8.1 Introduction 322
8.2 Associated equations 324
8.2.1 Associated Schrödinger equations 324
8.2.2 Dirac-like equation 324
8.2.3 Rigid rotator equation 325
8.3 Some properties of the spin equation 326
8.3.1 The inverse problem 326
8.3.2 General solution 327
8.3.3 Stationary solutions 328
8.3.4 Reduction of the external field 328
8.3.5 Transformation matrix 330
8.3.6 Evolution operator 331
8.4 Self-adjoint spin equation 332
8.4.1 General solution and inverse problem 332
8.4.2 Hamiltonian and Lagrangian forms of self-adjoint spin equation 333
8.5 Exact solutions of spin equation 334
8.6 Darboux transformation for spin equation 340
9 One-dimensional Schrödinger equation and its solutions 344
9.1 ESP I : V(x) = cx 345
9.2 ESP II : V(x) = V1x2 + V2x 346
9.3 ESP III : V(x) = -V1/x + V2/x2 346
9.4 ESP IV: V(x) = V1/x2 + V2x2 349
9.5 ESP V: V(x) = V1x-2cx + V2e-cx 350
9.6 ESP VI : V(x) = V1/sin2 cx + V2/cos2 cx 352
9.7 ESP VII : V(x) = V1 tan2 cx + V2 tan cx 354
9.8 ESP VIII : V(x) = V1 tanh2 cx + V2 tanh cx 355
9.9 ESP IX : V(x) = V1 coth2 cx + V2 coth cx 357
9.10 ESP X : V(x) = (V1 + V2 cosh 2x)/(sinh2 2x) 357
9.11 ESP XI : V(x) = (V1 + V2 sinh cx)/(cosh2 cx) 358
10 Coherent states 360
10.1 Introduction 360
10.2 Coherent states of the Heisenberg–Weyl group 361
10.2.1 HW algebra and HW group 361
10.2.2 CS of the HW group and Glauber CS 363
10.2.3 Heisenberg uncertainty relation and CS 365
10.2.4 Schrödinger–Glauber CS of a harmonic oscillator 367
10.3 Coherent states for systems with quadratic Hamiltonians 368
10.3.1 Basic equations 369
10.3.2 Integrals of motion linear in canonical operators q^ and p^ 370
10.3.3 Time dependent generalized CS 371
10.3.4 Standard deviations and uncertainty relations 373
10.3.5 Simple examples 374
A. Appendix 1 379
A.1 Pauli matrices 379
A.1.1 General properties 379
A.1.2 Vectors and spinors associated with Pauli matrices 381
A.1.3 Eigenvalue problem in space of complex spinors 383
A.1.4 Calculations of matrix elements 385
A.2 Dirac gamma-matrices 386
A.2.1 General properties 386
A.2.2 Gamma-matrix structure of the Lorentz transformation 389
B. Appendix 2 392
B.1 Laguerre functions 392
B.2 Hermite polynomials and Hermite functions 417
Bibliography 425
Index 441

Produkt-Details

• Erscheint lt. Verlag: 20.8.2014
• Reihe/Serie: De Gruyter Studies in Mathematical Physics
• Reihe/Serie: De Gruyter Studies in Mathematical Physics
• Reihe/Serie: ISSN
• Reihe/Serie: ISSN
• Zusatzinfo: 20 b/w ill.
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik
• Schlagworte: Dirac equation • Dirac Gleichung • Quantenelektrodynamik • quantum electrodynamics • Relativistic Quantum Mechanics • Relativistische Quantenmechanik
• ISBN-10: 3-11-037775-6 / 3110377756
• ISBN-13: 978-3-11-037775-0 / 9783110377750