Exact Solutions of Relativistic Wave Equations

Inhaltsverzeichnis

• 1. General properties of solutions of the Klein-Gordon and Dirac equations.
• 1 Basic notations and equations.
• 2 Some properties of the Dirac matrices, scalar product of relativistic wave functions and transformation laws.
• 3 On relations between solutions of the Dirac and Klein-Gordon equations.
• 4 Spin operators for the Dirac equation.
• 2. Most important exactly solvable problems in relativistic quantum mechanics.
• 5 Free relativistic particle. §1.
• 6 Motion of charge in a constant and homogeneous magnetic field. §2.
• 7 Motion of charge in the plane-wave electromagnetic field. §3.
• 8 Solutions in spherically symmetric fields. §4.
• 3. Exact solutions of relativistic equations of motion in external fields of special configuration.
• 9 General properties of motion of an electron in a crossed electromagnetic field.
• 10 Electron in crossed stationary electromagnetic fields. §§5 – 7.
• 11 Exact solutions for a charge in nonstationary crossed fields. §§8 – 19.
• 12 Charge motion in longitudinal electromagnetic fields. §§20 – 25.
• 13 Electron motion in a combination of crossed and longitudinal fields. §§26–39.
• 14 Exact solutions in the fields of nonstandard structure. §§40 – 43.
• 4. Green’s functions of relativistic wave equations in external electromagnetic fields.
• 15 Introduction.
• 16 Proper-time solution of equations for the Green’s functions in a constant field and in a combination of the latter with a plane-wave field.
• 17 Solutions and evolution functions of the Klein-Gordon and Dirac equations in a constant field and in its combination with a plane-wave field.
• 5. Coherent states of relativistic particles.
• 18 Introduction.
• 19 Coherent states of relativistic particles that move in an external electromagnetic field.
• 6. Charged particles in aquantized plane-wave field.
• 20 Basic equations and their interpretation.
• 21 Separation of the spin variables in the Dirac equation.
• 22 Separation of the spatial and photon variables. Conserved energy-momentum operator of the system.
• 23 Charge in quantized field of a monochromatic plane wave with arbitrary polarization.
• 24 Decomposition in linear polarizations for a quantized wave with arbitrary spectral composition and polarization.
• 25 Canonical forms of the Hamiltonian for quantized wave of general form.
• 26 Coordinate representation. Reduction of the Hamiltonian to the canonical form in the coordinate representation.
• 27 Stationary states.
• 28 Relations of orthogonality, normalization and completeness.
• 29 Integrals of motion. Coherent states.
• 30 Relation between the Volkov solutions and the wave functions in a quantized plane-wave field.
• 31 The Green’s functions of the Klein-Gordon and Dirac equations in a quantized plane-wave field.
• 32 Charge in the quantized plane-wave field combined with a classical electromagnetic field.
• 7. Exact solutions of Dirac-Pauli equations.
• 33 The Dirac-Pauli equation.
• 34 A charged particle with the anomalous moments in a constant and homogeneous magnetic field.
• 35 Exact solutions of the Dirac-Pauli equation for a particle in a plane electromagnetic wave field.
• 36 Solution of the Dirac-Pauli equation in a plane-wave field combined with a longitudinal running electric field.
• Appendix I Exact solutions of the one-dimensional stationary Schrödinger equation.
• Appendix II Systems with quadratic Hamiltonians.
• References.