Topological Effects in Quantum Mechanics

1 Introduction.- 2 Vector Potentials of Static Solenoids.- 21 Cylindrical Solenoids.- 2.2 Vector Potential of the Toroidal Solenoid.- 3 Electromagnetic Properties of Static Solenoids.- 3.1 Conditions for the Existence of Static Magnetic Solenoids.- 3.2 Currents, Magnetic Dipoles and Monopoles.- 3.3 Electric Static Solenoids.- 3.4 On the Disappearance of the Magnetic Field Inside a Solenoid.- 3.5 Vector Potentials of Toroidal Solenoids with Different Asymptotic Behaviour.- 3.6 Solenoids with Non-trivial Helicity.- 3.7 Electromagnetic Field of a Cylindrical Solenoid Moving Uniformly in a Medium.- 3.8 Generalised Helmholtz Coils and All That.- 4 Interaction of Magnetisations with an External Electromagnetic Field and a Generalisation of AmpÈRe’s Hypothesis.- 4.1 Introduction.- 4.2 Magnetisation, Toroidalisation, and a Generalisation of Ampère’s Hypothesis.- 4.3 Interaction with an External Electromagnetic Field.- 4.4 Magnetisations and the Debye Potential Representation.- 4.5 Physical Meaning of the ? Functions.- 4.6 Transition to Point-Like Sources.- 4.7 The Interaction of Charge Densities with an External Field.- 4.8 Motion of a Toroidal Solenoid in an External Electromagnetic Field.- 4.9 On the Inversion of the Debye Parametrisation.- 4.10 Concluding Remarks.- 5 Electromagnetic Properties of Time-Dependent Solenoids.- 5.1 Cylindrical Solenoids.- 5.2 Toroidal Solenoids.- 5.3 Elementary Time-Dependent Toroidal Sources.- 5.4 Vector Solutions of the Laplace Equation.- 6 Radiationless Time-Dependent Charge-Current Sources.- 6.1 Introduction.- 6.2 On Radiationless Topologically Non-Trivial Sources of Electromagnetic Fields.- 6.3 On Current Configurations Generating a Static Electric Field.- 6.4 On Electric Vector Potentials.- 6.5 More General Radiationless Sources.-6.6 Concluding Remarks Toroidal Radiationless Sources.- 7 Selected Topological Effects of Quantum Mechanics.- 7.1 An Elementary Exposition of the Bohm–Aharonov Effect.- 7.2 Attempts at Proving the Non-Existence of the B–A Effect.- 7.3 Scattering of Charged Particles on Two Infinite Cylindrical Solenoids.- 7.4 Scattering of Charged Particles by a Toroidal Solenoid.- 7.5 On the Super-Current Arising in a Superconducting Ring.- 7.6 The Time-Dependent Bohm–Aharonov Effect.- 7.7 Scattering of Charged Particles in an Ideal Multiply Connected Spacial Region.- 7.8 On the Alternative Interpretation of the Bohm–Aharonov Effect.- 7.9 The B–A Effect for Bound States.- 7.10 Quantum Impenetrability and Bohm–Aharonov Scattering.- 7.11 The Aharonov-Casher Effect.- 8 Topological Effects for the Free Electromagnetic Field.- 8.1 Introduction.- 8.2 Relativistic Helicity and Its Physical Meaning.- 8.3 Gauge-Invariant Representation for the Energy of a Weak Gravitational Field.- 8.4 Photon Densities.- 8.5 Numerical Results.- 8.6 On the Ehrenfest–Pauli Objections.- 8.7 On the Hegerfeldt Theorem and All That.- 8.8 Electromagnetic Waves versus Photons.- 8.9 Discussion.- 9 Topology of the Vavilov-Cherenkov Radiation Field.- 9.1 Introduction.- 9.2 Mathematical Preliminaries.- 9.3 Electromagnetic Field of a Uniformly Moving Charge on Finite and Infinite Intervals.- 9.4 Vavilov–Cherenkov Effect and Accelerated Motion of a Charged Particle.- References.