Interaction of molecules with electromagnetic fields. I. Classical particles and fields
Authors: Yang Kuo-Ho (Tom) , Hirschfelder Joseph O. 
Published: February 2003

Classical models of electrons and nuclei are developed which behave surprisingly like their quantum mechanical counterparts. The particles are finite in size and have both electrical charge density and current density. Classical spin results from the angular momentum of the particles rotating about their ’’center of mass’’. Internal electromagnetic fields are produced by the motion of the charged particles. These internal fields have dynamical properties of their own and produce the interaction between the particles. In this paper, the internal fields are derived through the second order of retardation or terms in 1/c2. Since higher order terms in the internal fields are not included, no difficulties are encountered with respect to the classical particles radiating. This same result could also be obtained by using equal parts of the retarded and advanced potentials. The Lagrangian, the (canonical, mechanical, and angular) momenta, the energy, and the Hamiltonian are derived in a gauge invariant manner for a ’’molecular system’’ of particles in the presence of an arbitrary (time and space varying) external electromagnetic field. The classical Hamiltonian corresponds term-by-term with the comparable quantum mechanical Breit–Pauli Hamiltonian. The correspondence includes the spin–orbit interaction terms, the Darwin terms (usually attributed to zitterbewegung), and the nuclear quadrupole terms (usually omitted from molecular Hamiltonians). A detailed comparison of classical and quantum mechanical dynamical properties will be given in a subsequent paper. Delta functions occur in both the classical and the quantum mechanical Hamiltonians to approximate the effect of overlapping when particles come close together. Otherwise, overlapping is generally not considered in either the classical or quantum treatments. In deriving the kinetic energy of a particle rotating about its center of mass, there was discovered a new relation which the rate of change of intrinsic spin (in the laboratory reference frame) must satisfy in order that the Lagrangian should completely describe the motion of the particle. This led to the discovery of a condition of compatibility between the dynamical properties of classical and quantum mechanical systems. This condition then led to a novel derivation of the relationship between the explicit expressions for the multipole moments of a particle in its laboratory reference frame (the observed moments) and the multipole moments in the particle’s rest frame (the intrinsic moments). It is assumed that an electron has only an intrinsic magnetic dipole moment whereas a nucleus has both an intrinsic magnetic dipole moment and an intrinsic electric quadrupole moment. The condition of compatibility also led to the derivation of canonical equations of change and generalized Poisson brackets, in both of which the spins are included as basic dynamical variables. The generalized Poisson brackets of the positions, the canonical momenta, and the spins exhibit the same forms as the commutation relations of their quantum mechanical counterparts. The (time and space varying) functions of the fields are expanded in Taylor series which have many of the features of Cartesian tensorial representations. If these functions were expanded in terms of spherical harmonics, the formulation would have been much more complicated.