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Annals of the New York Academy of Sciences

Magnetic Monopoles and Dipoles in Quantum Mechanicsa

Harry J. Lipkin, Murray Peshkin

Published Dec 1, 1986• 3 citations
Publisher

Abstract

The force on and the energy of a ''di-monopole'', which is the limiting case of a dipole made from two monopoles at zero separation and finite magnetic moment, interacting with an externally fixed magnetic field resulting from an electric current, is considered. A model involving only a monopole is used to illustrate the physical principles involved when magnetic sources move in a solenoidal field whose source is an electric current. The problems encountered in Hamiltonian theory are discussed. 5 refs., 3 figs. (LEW)

DOI
10.1111/j.1749-6632.1986.tb12424.x

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The force on and the energy of a ''di-monopole'', which is the limiting case of a dipole made from two monopoles at zero separation and finite magnetic moment, interacting with an externally fixed magnetic field resulting from an electric current, is considered. A model involving only a monopole is used to illustrate the physical principles involved when magnetic sources move in a solenoidal field whose source is an electric current. The problems encountered in Hamiltonian theory are discussed. 5 refs., 3 figs. (LEW)
In this work, we study the geometric phase and the dynamics of electric and magnetic dipoles. We present your origins, some relevant works in this area as well as your results. The appearance of a geometric phase is explained when a study of quantum dynamics of the particle is made, during a complete cycle and that is subject to cyclic adiabatic processes. The present study begins with an introduction about the geometric phase, where we present examples used to illustrate it. Next, the adiabatic theorem was presented in both classical and quantum systems and a discourse on adiabatic invariants. After this, we present its final mathematical formulation and also its obtaining for a spin 1/2 particle. Soon after, we obtained the Lagrangian and its Hamiltonian, we find the quantum phase that the particle acquired in the efect Aharonov- Bohm and Aharonov - Casher. We have also seen the quantum dynamics of both electric and magnetic dipoles, where we start from the Lagrangian and study its field configurations, we reach the Ananda phase, where in the next step we obtain phases for some of the Field configurations in the Ananda phase from phase of Aharonov - Casher where we made the electric dipole moment equal to zero, as well as the He - Mckellar - Welkens phase with magnetic dipole moment equal to zero, and the Casella efect thus obtaining a particle case of the Ananda phase.
Almost everything we know about atomic energy levels comes from the spectra resulting from radiative transitions between these levels. It is possible that the electric dipole moment of a charge distribution may vanish, in which case the radiation depends on the higher moments. The source of magnetic dipole radiation is a circulating current that reverses direction harmonically, as in a loop antenna. Since a steady-current loop can be regarded as equivalent to a static magnetic dipole, the oscillating current is equivalent to an oscillating dipole. If the electric dipole radiation is impossible, but magnetic dipole radiation can occur, the radiative process would be classed as "forbidden" because of its low intrinsic intensity. An oscillating linear dipole emits no radiation along the direction of its axis of vibration. Although the physical origin of magnetic moments is circulating currents, the effect of an oscillating circular current can be calculated by considering an equivalent linear magnetic dipole.
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.
The nonrelativistic quantum theory of a particle having both electric and magnetic charges moving in an arbitrary external electromagnetic field is presented. The theory is based on the hydrodynamic formulation of quantum mechanics. Dirac's quantization condition for the electric and magnetic charges is rederived as a consistency condition for the motion of the probability fluid. Neither the wave function nor the electromagnetic potential, which were the source of ambiguities in all other formulations, appears in our approach. Nevertheless, this theory has all the essential features of the standard quantum mechanics, including the superposition principle.