Abstract A canonical transformation is performed on the conventional Hamiltonian for the electromagnetic radiation field and an assemblage of neutral molecules in interaction. The new Hamiltonian has interaction energies expressed in terms of the electromagnetic fields alone and these energies have direct physical significance. The terms linear in e are the multipole interactions, both electric and magnetic, and the term quadratic in e is a generalization of the elementary diamagnetic energy shift. The intermolecular Coulomb energies have cancelled with transverse polarization fields in the new Hamiltonian, although the intramolecular Coulomb potentials are left unaffected. The equations of motion that follow from the new Hamiltonian are deduced. They are the so-called atomic-field equations for the Maxwell fields and Schrödinger equation for an electron wave in a transverse electromagnetic field. The former are the microscopic analogues of Maxwell’s equations in a medium (not restricted to dipole polarization fields) and the latter are dependent on the field strengths alone (not explicit functions of the vector potential).
Abstract A canonical transformation is performed on the conventional Hamiltonian for the electromagnetic radiation field and an assemblage of neutral molecules in interaction. The new Hamiltonian has interaction energies expressed in terms of the electromagnetic fields alone and these energies have direct physical significance. The terms linear in e are the multipole interactions, both electric and magnetic, and the term quadratic in e is a generalization of the elementary diamagnetic energy shift. The intermolecular Coulomb energies have cancelled with transverse polarization fields in the new Hamiltonian, although the intramolecular Coulomb potentials are left unaffected. The equations of motion that follow from the new Hamiltonian are deduced. They are the so-called atomic-field equations for the Maxwell fields and Schrödinger equation for an electron wave in a transverse electromagnetic field. The former are the microscopic analogues of Maxwell’s equations in a medium (not restricted to dipole polarization fields) and the latter are dependent on the field strengths alone (not explicit functions of the vector potential).
A recent investigation of the interaction between the Maxwell radiation field and neutral atoms or molecules has led us to consider the relation between the straightforward interaction Hamiltonian between particle (k) and the radiation field, and the more physical, and explicitly gauge invariant, couplings of the form q. E and µ. H, etc., where q and µ are the electric and magnetic dipole moments of a system of particles. That these latter form of interaction energies follow from well-formulated approximations for atoms in external fields has been known for many years (Goeppert-Mayer(1); see also Rosenfeld(2)). When the Maxwell field is an internal and dynamical field further discussion is necessary, especially concerning the Coulomb interactions. Their advantage has recently been shown in dealing with dipole interactions in the long range Van der Waals force between two neutral systems where the complete coupling, in dipole approximation, is given by — q.ET for each neutral system, where ET is the transverse electric field. The short-range London limit is contained in this coupling despite the absence of longitudinal electric fields (Power and Zienau (3)). In this note we extend these arguments to higher electric multipoles and magnetic multipoles. We also demonstrate their use in the quantum theory of scattering of light, and in particular the derivations of Stephen (4) of the phase angle between emergent beams in optical double refraction phenomena are shown to follow immediately.
Abstract As well‐known the dynamical interaction theory of molecules or atoms with the electromagnetic field may be based alternatively on one of two HAMILTONians, IH or IIH. where the vector potential appears in the interaction term of IH while this of IIH is expressed in terms of the electric and the magnetic field strength coupled to atomic multipoles. Using a calculus of transversal functional derivation with respect to a vector function we show that we can go from IH to IIH by a comperatively simple canonical transformation which may be interpreted within the classical theory as well as within as the quantized theory. The equivalence of both HAMILTONians is lost if, as customary in quantum optics, only a few atomic levels are taken into account. We give a plausible argument that in this case the multipole HAMILTONian IIH is more suitable.
Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.
Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.
Abstract The interaction between atoms or molecules and electromagnetic (EM) fields underlies all spectroscopic techniques and a great variety of desirable molecular properties. EM fields and EM waves are introduced via the famous Maxwell equations. The scalar and vector potential are defined, and the gauge freedom is outlined. The Coulomb gauge is adopted. The classical Hamiltonian for a charged particle in an EM field is derived, and from this the ‘minimal substitution’ rules for incorporating the fields in the quantum Hamiltonian are obtained. The operators describing the interaction of an electron with static electric and magnetic fields, including the magnetic fields from nuclear spins, are derived, followed by the derivation of the interaction between an electron and an EM wave.
This article deals with the fundamental problem of light-matter interaction in the quantum theory. Although it is described through the vector potential in quantum electrodynamics, it is believed by some that a hamiltonian involving only the electric and the magnetic fields is preferable. In the literature this hamiltonian is known as the Power-Zienau-Woolley hamiltonian. We question its validity and show that it is not equivalent to the minimal-coupling hamiltonian. In this article, we show that these two hamiltonians are not connected through a gauge transformation. We find that the gauge is not fixed in the Power-Zienau-Woolley hamiltonian. The interaction term is written in one gauge whereas the rest of the hamiltonian is written in another gauge. The Power-Zienau-Woolley hamiltonian and the minimal-coupling one are related through a unitary transformation that does not fulfill the gauge fixing constraints. Consequently, they predict different physical results. In this letter, we provide the correct quantum theory in the multipolar gauge with a hamiltonian involving only the physical fields.