Exact Solutions of Relativistic Wave Equations

ISBN-13: 9789401073264 / Angielski / Miękka / 2012 / 324 str.

V.G. Bagrov;D. Gitman

Opis

'Et moi * ...si favait su comment en revenir. One service mathematics bllS rendered the je n'y serais point aile.' human race. It hal put common sense back Jules Verne where it bdongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense', able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thOUght. A highly necessary tool in a world where both feedback and non- Iinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Spis treści

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 a quantized 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.