Breit equation

Abstract: A complete hamiltonian for a translating, rotating, vibrating molecule in the presence of a constant external electromagnetic field is derived starting with the Breit equation reduced to a non-relativistic form, correct to order c -2. A number of new terms appear, which have not been obtained in less precise derivations. These include mass polarization corrections to the orbital Zeeman, spin-orbit and orbit-orbit interactions together with spin-vibration, orbit-vibration and orbit-rotation interactions. In addition there is a vibrational Zeeman interaction which in certain circumstances may be of the same order of magnitude as rotational Zeeman terms. The final hamiltonian provides a starting point for future investigations of the possible effects of these new interactions on the microwave and radiofrequency spectra of both open and closed-shell molecules.

Abstract: 1 Non-relativistic Quantum Mechanics.- 1.1 Formal quantum mechanics.- 1.2 The Schrodinger equation.- 1.3 Heisenberg's uncertainty principle and related topics.- 1.4 Angular momentum.- 1.5 Electron spin.- 1.6 The need for a relativistic theory.- 2 Vector and Matrix Algebra.- 2.1 Vectors and vector multiplication.- 2.2 The repeated subscript convention for summation.- 2.3 The Kronecker delta ?ij.- 2.4 The ?ijk notation.- 2.5 The ?ijk sum rules.- 2.6 Examples I.- 2.7 The vector operator ?.- 2.8 The gradient.- 2.9 The divergence.- 2.10 The curl.- 2.11 Examples II.- 2.12 Second derivatives in vector calculus.- 2.13 The Dirac delta function.- 2.14 Matrices and determinants: a summary.- 2.15 Vectors in four dimensions.- 3 Classical Mechanics.- 3.1 Inertial frames and Galileo's relativity principle.- 3.2 The principle of least action.- 3.3 Lagrange's equations of motion.- 3.4 The Lagrangian for a system of particles.- 3.5 Constants of motion.- 3.6 The Hamiltonian.- 4 Special Relativity.- 4.1 Einstein's principle of relativity.- 4.2 The interval.- 4.3 The Lorentz transformation.- 4.4 Contraction, dilation and paradoxes.- 4.5 The transformation of velocities.- 4.6 The relativistic mechanics of a free particle.- 4.7 Four-vectors.- 5 The Interaction of Charged Particles with Electromagnetic Fields.- 5.1 Units.- 5.2 The electromagnetic potentials.- 5.3 The field vectors.- 5.4 The Lorentz transformation of electric and magnetic fields.- 5.5 Gauge transformations.- 5.6 Maxwell's equations.- 5.7 The potentials and fields due to a stationary charge.- 5.8 The potentials due to a moving charge.- 5.9 The interaction of two charged particles.- 5.10 The Thomas precession.- 6 The Classical Theory of Electromagnetic Fields.- 6.1 Continuous mechanical systems.- 6.2 The Lagrangian density for an electromagnetic field.- 6.3 The current four-vector.- 6.4 The second pair of Maxwell's equations.- 6.5 Electromagnetic waves.- 6.6 Solution of the wave equation for free space.- 6.7 The characteristic vibrations of an electromagnetic field.- 7 Relativistic Wave Equations.- 7.1 Quantization of classical equations.- 7.2 Gauge invariance of quantum mechanical equations.- 7.3 The Klein-Gordon equation.- 8 The Dirac Equation.- 8.1 The Dirac equation for a free electron.- 8.2 The Dirac operators ? and ?.- 8.3 The introduction of an electromagnetic field.- 8.4 Electron spin.- 8.5 Lorentz invariance of the Dirac equation.- 8.6 The negative energy solutions - positrons.- 8.7 The non-relativistic approximation of the Dirac equation.- 8.8 The method of small components.- 8.9 The Foldy-Wouthuysen transformation.- 8.10 The free electron.- 9 The Wave Equation for Many Electrons.- 9.1 The electromagnetic potentials due to a moving electron.- 9.2 The Hamiltonian for two electrons.- 9.3 The Breit equation.- 9.4 Reduction of the Breit equation to non-relativistic form.- 9.5 Radiative corrections.- 9.6 The many-electron Hamiltonian.- 10 The Molecular Hamiltonian.- 10.1 The introduction of nuclei.- 10.2 Finite nuclear size effects.- 10.3 Spectroscopically useful Hamiltonians.- 10.4 Effective Hamiltonians.- 11 The Hydrogen Atom.- 11.1 Non-relativistic theory for a one-electron atom.- 11.2 The non-relativistic approximation of the Dirac equation.- 11.3 The simultaneous eigenfunctions of j2, jz, l2, s2 and K.- 11.4 Commutation relations for the Dirac Hamiltonian.- 11.5 The Dirac equation in polar coordinates.- 11.6 Solution of the radial equations.- 11.7 The energy levels.- 11.8 Comparison of Dirac and non-relativistic atomic orbitals.- 11.9 The Lamb shift.- 11.10 More complicated systems.- 12 Quantum Field Theory.- 12.1 Quantization of the electromagnetic field.- 12.2 Solution of the one-dimensional harmonic oscillator equation.- 12.3 Creation and annihilation operators.- 12.4 Photons.- 12.5 Zero-point energy and vacuum fluctuations.- 12.6 Fermions and second quantization.- 13 The Interaction of Radiation and Matter.- 13.1 The interaction Hamiltonian.- 13.2 Time-dependent perturbation theory.- 13.3 Matrix elements of the interaction Hamiltonian.- 13.4 Absorption and emission.- 13.5 Comparison of the semiclassical and quantized theories.- 13.6 Multi-photon processes.- 13.7 The scattering of photons by molecules.- 13.8 Line widths and resonance fluorescence.- Appendix A Units.- A.1 SI units.- A.2 Conversion from the mixed (Gaussian) CGS system to the SI system.- A.3 Recommended values of physical constants.- Appendix B Vector Relations in Three Dimensions.- Appendix C General Bibliography.- Author Index.

Abstract: G.Breit's original paper of 1929 postulates the Breit equation as a correction to an earlier defective equation due to Eddington and Gaunt, containing a form of interaction suggested by Heisenberg and Pauli. We observe that manifestly covariant electromagnetic Two-Body Dirac equations previously obtained by us in the framework of Relativistic Constraint Mechanics reproduce the spectral results of the Breit equation but through an interaction structure that contains that of Eddington and Gaunt. By repeating for our equation the analysis that Breit used to demonstrate the superiority of his equation to that of Eddington and Gaunt, we show that the historically unfamiliar interaction structures of Two-Body Dirac equations (in Breit-like form) are just what is needed to correct the covariant Eddington Gaunt equation without resorting to Breit's version of retardation.