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The Dirac Equation and its Solutions Dirac Equation and its Solutions

:	Vladislav G. Bagrov, Dmitry Gitman
:	The Dirac Equation and its Solutions Dirac Equation and its Solutions
:	Walter de Gruyter GmbH& Co.KG
:	9783110377750
:	De Gruyter Studies in Mathematical PhysicsISSN
:	1
:	CHF 168.50
:

:	Theoretische Physik
:	English

:	441
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	ePUB/PDF

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.

	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