The Interactions of molecules with the electromagnetic field - Woolley, R; Woolley, R.G. (1970)
﻿ The way in which the macroscopic description of the interaction of the electromagnetic field with matter is related to the microscopic viewpoint has been known for many years. Two distinct steps are necessary to establish the connection. Firstly one has to develop a descriptive scheme that accounts for the dynamical behaviour of microscopic systems qua matter under the influence of the field which involves the construction of a model of matter, and secondly there is the problem of calculating the averages over a large number of microsystems of the quantities predicted by the dynamical scheme. The latter problem is essentially one of statistical mechanics. This thesis is concerned with the first aspect of the problem, that is the construction of a dynamical scheme to describe the interaction of the electromagnetic field with matter which we view as being composed of spatially separated aggregates of point charges called 'molecular systems'. The theory is illustrated with some examples of the type of calculations that are possible within this scheme. We make no attempt to discuss the associated problems in statistical mechanics. The conventional theory of the interaction of radiation with molecules has been standardized for many years. There are however some unsatisfactory features about it which very largely stem from the problems associated with the invariance of the hamiltonian under gauge transformations and the consequent freedom of choice of gauge. As shown by Power and Zienau (Phil. Trans. Roy. Soc. 1959, A251, 427) there exists a unique unitary transformation which eliminates the difficulties and casts the hamiltonian into a form in which the molecular multipoles are explicitly displayed. They also showed the extent to which earlier versions of the multipole hamiltonian were inexact about the definition of the fields involved. Power and Zienau however considered only dipolar and quadrupolar terms explicitly and it is desirable to extend the theory to encompass multipoles of arbitrary order. Moreover a proper account of the functional dependence of the canonical variables implied by the existence of the gauge transformations seems to be lacking in the case of molecular quantum electrodynamics. The first part of the thesis therefore is concerned with the problem of deriving in a consistent non-relativistic approximation the modified hamiltonian for general electric and magnetic multipolar interactions with the electromagnetic field. We start from the microscopic Maxwell-Lorentz theory of charged particles which we develop in a form that is in close analogy to the macroscopic theory of dielectrics. We then postulate a lagrangian from which the field and particle equations of motion may be recovered under suitable assumptions using the calculus of variations. This procedure however forces us to introduce the electromagnetic field 4-potential and thus there are more dynamical variables than degrees of freedom. To proceed to the hamiltonian therefore requires a discussion of the singular (degenerate) nature of the lagrangian and we use the theory first developed by Dirac (Can. J. Math. 1950, 2, 147) since this is the most direct way of performing calculations with lagrangians that exhibit degeneracy. Essentially the procedure consists of eliminating the dynamical variables that are not required in the account of the dynamical behaviour of the system, and when this is done the originally singular canonical scheme becomes regular as required. With a suitable lagrangian and the use of the Coulomb gauge condition to define a specific Lie algebra for the dynamical variables, the hamiltonian that finally emerges is the required generalization of the multipole hamiltonian. The importance of the Coulomb gauge condition is that it enables us to identify the canonical momentum with the transverse electric field strength operator to within a constant. The gauge condition is therefore an integral part of the theory. Having started from a non-relativistic classical lagrangian, the spin interaction terms are naturally absent from the hamiltonian and must be added, if required, in an ad hoc fashion. The second section of the thesis is concerned with the use of the theory within the context of perturbation theory based on an expansion of the resolvent operator of the complete hamiltonian. In Chapter 3 we give a detailed account of the calculation of the optical rotation angle of a molecule in terms of the properties of its constituent groups which we assume are electronically isolated. For simplicity at this stage the interaction potential is limited to the dipole approximation. Diagrammatic perturbation theory is used to simplify the calculation as much as possible. The final results of the calculation are in general agreement with those in the literature but the method seems to give a particularly clear account of the physical processes involved. In Chapter 4 we consider the complete multipole hamiltonian and examine the properties of its matrix elements in order to assess the feasibility of extending calculations in the dipole approximation to include higher multipoles. Some interesting new integral representations of the interaction terms are developed which may be useful in calculations. We conclude that there is no difficulty in generalizing calculations if the process in question involves real photons, but that in general 'virtual' photon processes may only be dealt with if a cut-off for the field energy is introduced. We note that if the theory is to be consistent with our original restriction to the non-relativistic case then a cut-off is required but we do not analyse such a theory. Finally in the Discussion, we attempt to assess our formulation of the multipole hamiltonian in relation to other versions and find agreement only with Power (introductory Quantum Electrodynamics, 1964) though of course our final hamiltonian is not limited to quadrupolar terms. By extending the analysis of the integral representation of the electric multipole term in the hamiltonian, which we developed in Chapter 4, we are able to give a fully quantum mechanical justification of the unitary transformation operator employed by Power and Zienau (1959) and Power (1964). Our argument follows from the recognition that the phase of probability amplitudes may be altered in the presence of an electromagnetic field ('Bohm-Aharonov' effect). In short the multipole hamiltonian is related to the conventional hamiltonian by a unitary operator that changes the phase of the state vectors of the two descriptions by an amount that corresponds to formally moving every particle of a molecular aggregate to the centre of mass of the aggregate (or vice versa to go from the multipole description to the conventional one) in the presence of an electromagnetic field described by a given vector potential. We conclude by discussing some of the difficulties inherent in the current theory of molecular quantum electrodynamics which appear to be of a fundamental nature since they arise from our manner of describing charged particles.