Edwin Albert Power

Coulomb Gauge in Non-Relativistic Quantum Electro-Dynamics and the Shape of Spectral Lines

An examination is made of the Green dyadic for the propagation functions of non-relativistic quantum electrodynamics in the Coulomb gauge div A = 0. The existence of false precursors is shown to be associated with singularities at the origin in certain momentum space representations of the propagators and this allows a prescription to be given to eliminate such precursors. This prescription is proved equivalent to an exact consideration of the near fields associated with quantum transitions. The analogous treatment for the state vectors in decaying systems is carried through and is applied in detail to the problems of the shape of natural spectral lines and the transition rate for stimulated emission. It is shown that the expected line shape for the Lamb 1057 Mc/s line in hydrogen is Lorentzian within the context of Lamb’s experiments—intensity against Zeeman splitting energy. This is in contradiction to the predictions of Arnous and Heitler. The distribution in k (circular frequency) for a natural line, on the other hand, has a factor ka times the resonant denominator. The definitive position between the conflicting transition rates in the literature is given. The relationship of the calculations to a gauge-independent Hamiltonian, considered in the paper, is investigated in detail. This Hamiltonian, transformed canonically from the conventional one, is shown to be valid for internal, dynamical, quantized electromagnetic fields; its analogue in the case of external applied fields being well known. It is demonstrated that the transformation eliminates many of the difficulties associated with the false near fields and that it also allows certain radiative effects, where the atom acts as a whole, to be computed in a straightforward manner.

Circular dichroism: A general theory based on quantum electrodynamics

A theory of circular dichroism is developed through a direct calculation of the absorption rates for circularly polarized light on molecules. From this a differential rate as between left and right circularly polarized light is calculated. This is immediately related to the experimental data on circular dichroism. When the calculations are restricted to the dipole approximation we reproduce the rotatory strength in the form of a differential Einstein B coefficient. Higher moments are considered in detail and their effects analyzed for both the locked‐in situation and the case of randomly oriented molecular absorbers. A general expression is obtained for the differential absorption rate.

The Interaction of Optically Active Molecules

The difference in interaction energies between two laevo molecules, and one dextro and one laevo, are analysed. Formulae are given for the discriminating terms in the dispersion and induction energies for molecules in fixed orientations, between freely rotating molecules, and molecules rotating about the intermolecular axis. Discrimination in the resonance interaction between optical isomers appears in the retarded interaction (in dipole approximation) as a term with an inverse square dependence on distance in the near zone. Its significance is discussed for pairwise interactions and in crystalline systems.

On the effects of diagonal dipole matrix elements in multi-photon resonance profiles using two-level systems as models

Solutions of the time-dependent Schrodinger equation, in the semi-classical electric dipole approximation, are developed for the interaction of a two-level system, possessing both transition and diagonal dipole matrix elements, with applied static and sinusoidal electric fields. Exact solutions can only be obtained subject to conditions involving the orientations and strengths of the applied fields. These, and perturbative extensions, are used to discuss some of the features of multi-photon processes that depend on the existence of non-zero diagonal dipole matrix elements. Some of the results apply, in the limit of vanishing permanent moments, to the atomic two-level problem.

The Long Range van der Waals Forces between Non-Identical Systems

The interaction energy between two dissimilar non-ionized molecules or atoms is calculated in fourth-order perturbation theory and dipole approximation. The interaction Hamiltonian involves the charge distribution with the complete Maxwell field and not only the Coulomb interaction between charges. At close separations between the two systems (still large compared with molecular diameters) the interaction energy is of course that corresponding to the London force. However, for separations large compared with the characteristic wavelengths associated with transitions within the molecules the London force is modified considerably. In the case of two molecules in the ground state this modification was first found by Casimir & Polder. If one of the molecules is in an excited state new effects appear at these large distances. The energy of interaction depends on the orientation of the transition moment in the excited molecule with respect to the vector displacement between the two systems. In both transverse and longitudinal orientations the potential law is considerably stronger than the R-7 of the ground state-ground state interaction. For transverse orientations there is an unmodulated R-2 energy dependence which though very weak individually could give rise to considerable effects when the excited molecule is in a macroscopic environment.

The non-additive dispersion energies for N molecules: a quantum electrodynamical theory

With the aid of the Heisenberg operators for the electromagnetic field in the neighbourhood of several molecules, the many-body interaction potentials are calculated by finding the energy of a test molecule in this field. In the first instance the calculation is made for the case where the intermolecular separations are very large. In this far-zone range, the energies depend on the molecules through their static polarizabilities. The three-body non-additive dispersion energy is calculated for an arbitrary configuration and the dependence on the geometry is investigated. The four-body non-additive potential is found for the regular tetrahedral configuration. The theory is extended to cover all separations outside regions of overlap. It is shown that the interaction energy in this case depends on the dynamic polarizabilities. The full Casimir-Polder potential follows directly from the general expression. The near-zone limit of the complete formula tends to the form obtained by using electrostatic couplings only: as with, for example, the Axilrod-Teller potential for N = 3. The work described here presents an alternative viewpoint to the conventional perturbation theory and lends further insight into the nature of intermolecular forces.

The multipolar Hamiltonian in radiation theory

The multipolar Hamiltonian is widely used in applications of quantum electrodynamics to quantum optics and theoretical chemistry. In this paper, it is shown how this form of the Hamiltonian, normally derived within the Coulomb gauge, may be obtained from a Lagrangian in an arbitrary gauge. The method involves the construction of the Routhian functional to eliminate ignorable coordinates from which the Hamiltonian is obtained. Further, the electrostatic interactions arise from a constraint and are independent of the initial choice of the scalar potential. The contributions from Röntgen currents, arising from dielectric motion, are allowed for by including nuclear motions in the theory.

On the Interaction of Elliptically Polarized Light with Molecules; the Effects of Both Permanent and Transition Multipole Moments on Multiphoton Absorption and Chiroptical Effects

Abstract The rates for the absorption of radiation by complex systems are calculated for the most general elliptically polarized light. Absorption is considered from intense beams where multiphoton processes are important so that the mechanism must be considered in the nonlinear regime. The results are presented through generalized B-coefficients, which are basically functions of the absorbing molecules, the coherence and intensity dependency of the rates being factored out. The present calculations extend our previous ones in so far as the emphasis is on the effects of the ellipticity of the light beam. Extensive results are given for two- and three-photon absorption rates for molecules with both permanent and transition electric dipole, magnetic dipole and electric quadrupole moments being effective. Special consideration is given to chiroptical effects, both where these are due to orientational dependence of the molecule with respect to the light beam and where these are due to the intrinsic structure ...

Optical Activity as a Two‐State Process

The phenomenon of optical activity is analyzed as a quantum two‐state process with transitions between photon states of perpendicular plane polarizations, but with the same momentum. The matrix element connecting the two states is computed using quantum electrodynamics. The Rosenfeld—Condon formula for the angle of optical rotation is derived without recourse to Maxwells equations. The relationship between refractive index and molecular polarizability is also derived using quantum electrodynamics. The theory is extended to obtain the difference between refractive indices of an optically active medium with respect to right and left circularly polarized light and the difference is related to optical rotation.

The Dynamic Terms in Induced Circular Dichroism

Circular dichroism can be induced in an achiral molecule both by the static and time-dependent fields of neighbouring chiral molecules. Here the time-dependent pairwise interactions are calculated by quantum electrodynamics for all separations of the interacting pair. It is shown how, by two alternative canonical transformations of the Hamiltonian, the full results can be recovered much more simply, with advantages also in physical insight. For molecule pairs in fixed relative orientations the leading term varies with distance as R-3. Various orientational averages are also calculated, applicable to pairs in the gas phase coupled weakly by intermolecular forces. For coupling either through permanent electric dipoles, or by the dispersion force, the leading term is in R-9; in the first the dependence on temperature is on T-2, in the second on T-1.