Contents -- Preface -- Acknowledgements -- 1 Introduction -- 1.1 Book content -- 1.2 Notation -- 2 Relativistic equations of motion -- 2.1 Classical equations -- 2.1.1 Maxwell equations -- 2.1.2 Equations of motion for a charge in an electromagnetic field -- 2.1.3 Hamilton-Jacobi equation -- 2.2 K-G equation -- 2.2.1 General -- 2.2.2 Evolution function and completeness relations -- 2.2.3 Hamiltonian forms of the K-G equation -- 2.3 Dirac equation -- 2.3.1 General -- 2.3.2 Evolution function and completeness relation -- 2.3.3 Reducing Dirac equation into two independent sets of second-order equations for spinors -- 2.3.4 Reducing Dirac equation into two independent sets of fourth-order equations for scalar functions -- 2.3.5 Squaring the Dirac equation -- 2.4 Spin operators -- 3 Basic exact solutions -- 3.1 Free particle motion -- 3.1.1 Classical motion -- 3.1.2 States with a given momentum -- 3.1.3 Positive and negative frequency solutions -- 3.1.4 Light-cone variables and coherent states -- 3.1.5 States with given angular momentum projection -- 3.2 Particles in plane-wave field -- 3.2.1 Plane-wave electromagnetic field -- 3.2.2 Classical motion in the plane-wave field -- 3.2.3 Quantum motion in plane-wave field -- 3.3 Particles in BGY field -- 3.3.1 BGY field -- 3.3.2 Classical motion in a BGY field -- 3.3.3 Quantum motion in a BGY field -- 3.4 Particles in a constant and uniform magnetic field -- 3.4.1 Introduction -- 3.4.2 Pageś and Rabiś solutions -- 3.4.3 Creation and annihilation operators -- 3.4.4 Stationary states -- 3.4.5 Orthonormality and completeness of stationary states -- 3.4.6 Coherent states -- 3.4.7 Zero magnetic field limit -- 3.4.8 Some other types of nonstationary states -- 3.5 Particles in spherically symmetric fields -- 3.5.1 General.

3.5.2 Separation of variables in K-G and Dirac equations -- 3.5.3 Specification of potentials and complete classical solution -- 3.5.4 Azimuthal motion -- 3.5.5 Radial motion -- 3.6 Particles in the Aharonov-Bohm field and in its superpositions with other fields -- 3.6.1 Introduction -- 3.6.2 Aharonov-Bohm field -- 3.6.3 Magnetic-solenoid field -- 3.6.4 Quasicoherent states in the magnetic-solenoid field -- 3.6.5 Aharonov-Bohm field and additional electromagnetic fields -- 4 Particles in fields of special structure -- 4.1 Introduction -- 4.2 Crossed electromagnetic fields -- 4.2.1 General -- 4.2.2 Stationary crossed fields -- 4.2.3 Nonstationary crossed fields -- 4.3 Longitudinal electromagnetic fields -- 4.3.1 General -- 4.3.2 Longitudinal motion in the electric field -- 4.3.3 Transversal motion in the magnetic field -- 4.4 Superposition of crossed and longitudinal fields -- 4.4.1 General -- 4.4.2 Crossed and longitudinal electric field -- 4.4.3 Crossed and longitudinal electric and magnetic fields -- 4.5 Fields of nonstandard structure -- 5 Dirac-Pauli equation and its solutions -- 5.1 Introduction -- 5.2 Constant and uniform magnetic field -- 5.3 Plane-wave field -- 5.4 Superposition of a plane-wave field and a parallel electric field -- 6 Propagators of relativistic particles -- 6.1 Introduction -- 6.2 Proper-time representations for particle propagators -- 6.2.1 General -- 6.2.2 Proper-time representations in a constant uniform field and a plane wave field -- 6.3 Path-integrals for particle propagators -- 6.3.1 Path integral for K-G propagator -- 6.3.2 Path integral for the Dirac propagator in even dimensions -- 6.3.3 Path integral for the Dirac propagator in odd dimensions -- 6.3.4 Classical and pseudoclassical description of relativistic particles.

6.4 Calculations of Dirac propagators using path integrals -- 6.4.1 Spin factor in 3 + 1 dimensions -- 6.4.2 Propagator in the constant uniform electromagnetic field -- 6.4.3 Propagator in a constant uniform field and a plane wave field -- 6.4.4 Propagator in a constant uniform field in 2 + 1 dimensions -- 7 Electron interacting with a quantized electromagnetic plane wave -- 7.1 Dirac equation with quantized plane wave -- 7.1.1 General -- 7.1.2 Separation of variables -- 7.2 Quantized monochromatic plane wave with arbitrary polarization -- 7.3 Quantized plane wave of general form -- 7.4 Canonical forms for Hamiltonian of quasiphotons -- 7.5 Stationary and coherent states -- 7.5.1 Stationary states -- 7.5.2 Relations of orthogonality, normalization and completeness -- 7.6 Reduction to Volkov solutions -- 7.7 Electron interacting with quantized plane-wave and with external electromagnetic background -- 7.7.1 Classical plane wave along the quantized field -- 7.7.2 Classical magnetic field directed along the quantized plane wave -- 7.8 Linear and quadratic combinations of creation and annihilation operators -- 7.8.1 Linear combinations -- 7.8.2 Quadratic combinations -- 8 Spin equation and its solutions -- 8.1 Introduction -- 8.2 Associated equations -- 8.2.1 Associated Schrödinger equations -- 8.2.2 Dirac-like equation -- 8.2.3 Rigid rotator equation -- 8.3 Some properties of the spin equation -- 8.3.1 The inverse problem -- 8.3.2 General solution -- 8.3.3 Stationary solutions -- 8.3.4 Reduction of the external field -- 8.3.5 Transformation matrix -- 8.3.6 Evolution operator -- 8.4 Self-adjoint spin equation -- 8.4.1 General solution and inverse problem -- 8.4.2 Hamiltonian and Lagrangian forms of self-adjoint spin equation -- 8.5 Exact solutions of spin equation.

8.6 Darboux transformation for spin equation -- 9 One-dimensional Schrödinger equation and its solutions -- 9.1 ESP I : V(x) = cx -- 9.2 ESP II : V(x) = V1x2 + V2x -- 9.3 ESP III : V(x) = -V1/x + V2/x2 -- 9.4 ESP IV: V(x) = V1/x2 + V2x2 -- 9.5 ESP V: V(x) = V1x-2cx + V2e-cx -- 9.6 ESP VI : V(x) = V1/sin2 cx + V2/cos2 cx -- 9.7 ESP VII : V(x) = V1 tan2 cx + V2 tan cx -- 9.8 ESP VIII : V(x) = V1 tanh2 cx + V2 tanh cx -- 9.9 ESP IX : V(x) = V1 coth2 cx + V2 coth cx -- 9.10 ESP X : V(x) = (V1 + V2 cosh 2x)/(sinh2 2x) -- 9.11 ESP XI : V(x) = (V1 + V2 sinh cx)/(cosh2 cx) -- 10 Coherent states -- 10.1 Introduction -- 10.2 Coherent states of the Heisenberg-Weyl group -- 10.2.1 HW algebra and HW group -- 10.2.2 CS of the HW group and Glauber CS -- 10.2.3 Heisenberg uncertainty relation and CS -- 10.2.4 Schrödinger-Glauber CS of a harmonic oscillator -- 10.3 Coherent states for systems with quadratic Hamiltonians -- 10.3.1 Basic equations -- 10.3.2 Integrals of motion linear in canonical operators q^ and p^ -- 10.3.3 Time dependent generalized CS -- 10.3.4 Standard deviations and uncertainty relations -- 10.3.5 Simple examples -- A. Appendix 1 -- A.1 Pauli matrices -- A.1.1 General properties -- A.1.2 Vectors and spinors associated with Pauli matrices -- A.1.3 Eigenvalue problem in space of complex spinors -- A.1.4 Calculations of matrix elements -- A.2 Dirac gamma-matrices -- A.2.1 General properties -- A.2.2 Gamma-matrix structure of the Lorentz transformation -- B. Appendix 2 -- B.1 Laguerre functions -- B.2 Hermite polynomials and Hermite functions -- Bibliography -- Index.