A physical interpretation is given for the Heisenberg equations implied by the usual phenomenological Hamiltonian for a neutral, magnetic dipole in a radiation field. If the Heisenberg equations that characterize the quantum description are to be formally the same as their classical counterparts, a distinction must be made between the canonical and orbital momenta, even for a neutral system. This leads to a novel interpretation of the spin-orbit energy and a reidentification of the electric polarizability of the system. The general ideas are illustrated by considering the exactly soluble special cases of a neutral magnetic dipole with a nonvanishing electric polarizability in uniform, static fields and in the field of a monochromatic, plane wave. A Lagrangian formalism corresponding to these ideas is also presented.
A physical interpretation is given for the Heisenberg equations implied by the usual phenomenological Hamiltonian for a neutral, magnetic dipole in a radiation field. If the Heisenberg equations that characterize the quantum description are to be formally the same as their classical counterparts, a distinction must be made between the canonical and orbital momenta, even for a neutral system. This leads to a novel interpretation of the spin-orbit energy and a reidentification of the electric polarizability of the system. The general ideas are illustrated by considering the exactly soluble special cases of a neutral magnetic dipole with a nonvanishing electric polarizability in uniform, static fields and in the field of a monochromatic, plane wave. A Lagrangian formalism corresponding to these ideas is also presented.
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 equations of motion and the class of Hamiltonians for electrodynamics, both classical and quantal, are considered for the interaction of radiation with bound charges within the electric-dipole approximation. It is emphasized that, despite the uniqueness of the equations of motion for physical variables, the form of the Hamiltonian is not unique and that a physical system is associated with an equivalence class of Hamiltonians. Two of the members of this class are the minimal-coupling and multipolar forms extensively used in quantum optics. It is shown that although different choices of Hamiltonians lead to different first-order equations of motion that connect canonical coordinates and momenta, they lead to the same second-order equations of motion for the physical variables and physical electromagnetic fields. In the quantum theory of electrodynamics the role played by the choice of representation for the operators and for the states is analyzed with care. The choice of representation leads to a freedom additional to that provided by canonical transformations in classical electrodyamics, although the two lead to the same operator equations of motion.
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 This paper is based on the idea that wave mechanics is primarily suitable for giving an indirect description of nature. The world to which the equations of the, theory here proposed are directly applicable is an unnatural one, containing sources and sinks of matter which give to its radiational laws a symmetry lacking in nature. The theory may be regarded as consisting of two postulates. The first (which may, if desired, be regarded as a definition) specifies the character and properties of an unnatural assembly of systems, each consisting of an electron in a static potential field U. It states that, of the energy radiated in an elementary solid angle of standard magnitude, the part which will pass through an analyzer which transmits the component of electric field in a given direction (perpendicular to the line of observation) is determined by applying the usual Schrödinger interpretations (of density, current and material energy) to a Schrödinger equation perturbed by a term which is a certain constant multiplied by the electric field, in that direction, acting on the electron, due to the potential field U. (For comparison with classical radiation theory, it will be noted that the perturbation term vanishes in any region in which, from the classical standpoint, the electron is not accelerated, in that direction, by the field U). In the process of specifying the radiation from the assembly, the Schrödinger interpretations will have specified also the nature and magnitude of the sources and sinks of electrons (each in a field U). Their presence is due, mathematically speaking, to the multiplying constant mentioned above being a complex quantity. From these results the properties of a natural assembly are specified by means of the second postulate, which states that when the sources and sinks are removed, any process which in their absence is meaningless or physically inconceivable will cease, and all other processes will remain unaffected. The calculation is readily extended to include the effect of an external field of radiation. The theory requires no representation of radiation other than the Maxwellian wave, and will be free from the difficulties encountered in other theories.
Light occupies a special position in our attempts to understand nature both classically and quantum mechanically. We recall that Newton, who made so many fundamental contributions to optics, championed a particle description of light and was not favorably disposed to the wave picture of light. However, the beautiful unification of electricity and magnetism achieved by Maxwell clearly showed that light was properly understood as the wave-like undulations of electric and magnetic fields propagating through space. The central role of light in marking the frontiers of physics continues on into the twentieth century with the ultraviolet catastrophe associated with black-body radiation on the one hand and the photoelectric effect on the other. Indeed, it was here that the era of quantum mechanics was initiated with Planck's introduction of the quantum of action that was necessary to explain the black-body radiation spectrum. The extension of these ideas led Einstein to explain the photoelectric effect, and to introduce the photon concept. It was, however, left to Dirac to combine the wave-and particlelike aspects of light so that the radiation field is capable of explaining all interference phenomena and yet shows the excitation of a specific atom located along a wave front absorbing one photon of energy. In this chapter, following Dirac, we associate each mode of the radiation field with a quantized simple harmonic oscillator, this is the essence of the quantum theory of radiation.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.