The Hamiltonian of a system of charged particles interacting with the electromagnetic field is investigated. For an arbitrary system the multipole expansion of the interaction between the system and the field is derived by means of a suitable canonical transformation. The transformed Hamiltonian is obtained from the Hamiltonian of the system by replacing the momenta by the transformed kinetic momenta and by adding to the Hamiltonian a term representing the interaction of the system with the electric component of the field. By expanding this interaction term, as well as the transformed momenta, in powers of the dimension of the system over the wavelength, the multipole expansion of the Hamiltonian is obtained. For a system interacting with a classical field the multipole form of the Hamiltonian is exactly equivalent to the original Hamiltonian. For a quantized field this is not true, and the multipole form of the transformed Hamiltonian is shown to be equivalent to the original Hamiltonian only for first-order radiation processes.
This Letter establishes the analytical structure of the dipole radiation transition operator, taking into account non-dipole effects for a multielectron atom. The results summarize those of Komninos Y. et al., Phys. Rev. A 86, 023420 (2012), obtained in an approximation of a "split" electron of a continuous spectrum. The result for the complete operator of a multipole radiation transition [see formula (10)] can be specified within the framework of the methods of the theory of irreducible tensor operators and for higher multipoles.
A compact and at the same time general tensor formalism adapted to the theoretical prediction of a wide range of non-linear processes due in general to multipole interactions between atomic or molecular systems and electromagnetic fields is proposed. Recurring to a semi-classical method, the expectation values of the 2 n -pole induced electric moment P e ( n ) and 2 n -pole magnetic moment P m ( n ) for a transition k → l are computed explicitly to within the third order of quantum-mechanical perturbation theory approximation. The formalism is demonstrated with regard to its utility and simplicity on the example of electric and magnetic multipole light scattering with multiharmonic frequencies ω kl + ω 1 + ω 2 +... + ω r .
A geometric theory of the irreducible tensor operators of quantum spin systems. It is based upon the Maxwell-Sylvester geometric representation of the multipolar electrostatic potential. In the latter, an order-
ℓ multipolar potential is represented by a collection of
ℓ equal length vectors, i.e. by
ℓ points on a sphere, instead of by its components on some fixed (but arbitrary) basis. The geometric representation offers a much more appropriate tool for getting physical insight on specific characteristics of a multipole, such as its symmetries, or its departure from ideal symmetry. We derive explicit expressions enabling to perform any calculations we may need to perform on multipoles. All relevant quantities are eventually expressed in terms of scalar products of pairs of vectors (i.e., in terms of geometric quantities such as lengths and angles). The whole formalism is entirely independent of any particular choice of coordinate, and needs no use of the somehow abstract formalism traditionally used when dealing with angular momenta. The formalism is then applied to treat the problem of the irreducible tensor operators of quantum spin systems. It enables to completely dispense with the calculation and use of the Stevens operators, which can be quite complicated even for moderate values of
ℓ. Explicit expressions for the calculation of expectations values of physical observables are derived. They essentially consist in combinations of scalar products of vector pairs. Together with the coherent state representation of the quantum states of spin systems, this provides a complete geometric, coordinate-free, description of the states, dynamics and physical properties of these systems.
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.