Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

Transverse current density in nonrelativistic quantum mechanics

Ole Keller

Published May 13, 2014• 0 citations
Publisher

Abstract

We have seen that the source domain of a transverse photon, a quantum physical object described in this book via the transverse part of the vector potential (AT (r, t)), may be identified with the transverse part of the charged-particle current density (JT (r, t)). In general, many-body quantum physics is needed to determine JT (r, t). However, it is possible to illustrate a number of important principles related to the characterization of the photon source domain by limiting oneself to a study of the single-particle (electron) current density concept in nonrelativistic quantum mechanics.

DOI
10.1201/b16917-34

Related papers

Maxwell Operator Equations. Einstein Causality

Ole Keller

Starting from the well-known plane-wave decomposition of the operator associ- ated with the transverse part of the vector potential (Aˆ T) , and by making use of the operator relations defining the relations between and magnetic (Bˆ [with ∇ · Bˆ = 0]) field operators, the plane-wave mode expansions for Eˆ T and Bˆ are established. Subsequently, the general operator relation ∇ × Eˆ T = −∂ Bˆ/ ∂t follows. From the Heisenberg equation of motion for the magnetic field operator the inhomogeneous Maxwell operator equation ∇ × Bˆ = µ 0 Jˆsym + c − 2 ∂ Eˆ T/∂t is established. Here the operator Jˆsym T is the symmetrized transverse current den- sity operator in the space-time domain. The transverse set of Maxwell operator equations are of utmost importance for the subsequent photon energy operator wave function formalism. For globally coherent quantum fields, where the par- ticle source current density distribution is quasi-classical, dynamical equations for the two helicity species of Riemann-Silberstien field operators are obtained. Neglect of the quantum field fluctuations in the coherent state makes the bridge to photon wave mechanics in the Oppenheimer-Riemann-Silberstein-Bialynicki approach. The chapter is closed with a discussion of response theory and Ein- stein causality.

CRC Press eBooks•Aug 6, 2024•0 citations

Currents as Coordinates in Nonrelativistic Quantum Mechanics

WILLIAM J. PARDEE, Leonard Schlessinger, Jon Wright

The formulation of nonrelativistic quantum mechanics using currents and densities as coordinates is investigated. A general solution for a single-particle theory is presented, and several many-body problems are discussed.

Physical Review•Nov 25, 1968•11 citations

On the Nonrelativistic Quantum-Mechanical Hamiltonian with $1/c^2$ terms. Transverse current-current interaction

L. Bányai

We extend the standard solid-state quantum mechanical Hamiltonian containing only Coulomb interactions between the charged particles by inclusion of $1/c^2$ terms representing (transverse) current-current interaction. For its derivation we use the classical formulation of Landau-Lifshitz, however consequently in the Coulomb gauge. Our Hamiltonian does not coincide with the Darwin Hamiltonian and we emphasize the mathematical inconsistency in its derivation. We show, that the quantized version of our Hamiltonian is equivalent to the non-relativistic QED considering only states without photons and retaining only terms of order $1/c^2$. The importance of this extended Hamiltonian lies in the possibility to distinguish external from internal magnetic fields. This aspect may be relevant for theories of the Meissner effect.

arXiv:1905.02390•May 7, 2019•1 citation

Space-time description of photon emission from an atom

Ole Keller

Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.

Physical Review A•Jul 18, 2000•44 citations

Propagator picture of the spatial confinement of quantized light emitted from an atom

Ole Keller

A field-quantized electromagnetic propagator description emphasizing the spatial localization of light emitted from a single-electron atom is presented. With the aim of obtaining a better insight into the quantum electrodynamics in the near-field zone of the atom, two approaches, formally different but equivalent in respect to their physical predictions, are studied. From the Maxwell equations, among the transverse operators, a transverse propagator picture in which the source region for the field is identified with the spatial domain occupied by the transverse current density of the atom emerges. In this picture the field propagation is isotropic and always retarded. By identifying the source region with that of the total atomic current, the transverse propagator becomes anisotropic, and attains a nonretarded self-field part and a near-field part different from zero only for spacelike events. By changing the Coulomb Lagrangian, a nonrelativistic Hamiltonian formalism adequate for reconciling the photon concept with the anisotropic propagator picture is introduced. In this formalism the transverse self-field energy is transferred to the particle Hamiltonian, and the interaction Hamiltonian now includes an interaction between the retarded field and the transverse self-field dynamics. The link between the standard (nonpropagator) theory, often used to argue that the total electromagnetic field is always retarded, and the transverse propagator theory is established. The equivalence between the two approaches is proved by demonstrating that the standard theory in fact includes a nonretarded response in the spatial domain occupied by the longitudinal part of the induced atomic current density.

Physical Review A•Nov 1, 1998•30 citations

Energy Density and Current in Quantum Theory

W. N. Mathews

The concepts of energy density and energy current density for a single particle in nonrelativistic quantum theory are discussed. It is shown that the energy density ε and the energy current density Q are related by the equation (δε/δt)+∇·Q = S, where S represents external power sources.

American Journal of Physics•Mar 1, 1974•10 citations
We have seen that the source domain of a transverse photon, a quantum physical object described in this book via the transverse part of the vector potential (AT (r, t)), may be identified with the transverse part of the charged-particle current density (JT (r, t)). In general, many-body quantum physics is needed to determine JT (r, t). However, it is possible to illustrate a number of important principles related to the characterization of the photon source domain by limiting oneself to a study of the single-particle (electron) current density concept in nonrelativistic quantum mechanics.
Starting from the well-known plane-wave decomposition of the operator associ- ated with the transverse part of the vector potential (Aˆ T) , and by making use of the operator relations defining the relations between and magnetic (Bˆ [with ∇ · Bˆ = 0]) field operators, the plane-wave mode expansions for Eˆ T and Bˆ are established. Subsequently, the general operator relation ∇ × Eˆ T = −∂ Bˆ/ ∂t follows. From the Heisenberg equation of motion for the magnetic field operator the inhomogeneous Maxwell operator equation ∇ × Bˆ = µ 0 Jˆsym + c − 2 ∂ Eˆ T/∂t is established. Here the operator Jˆsym T is the symmetrized transverse current den- sity operator in the space-time domain. The transverse set of Maxwell operator equations are of utmost importance for the subsequent photon energy operator wave function formalism. For globally coherent quantum fields, where the par- ticle source current density distribution is quasi-classical, dynamical equations for the two helicity species of Riemann-Silberstien field operators are obtained. Neglect of the quantum field fluctuations in the coherent state makes the bridge to photon wave mechanics in the Oppenheimer-Riemann-Silberstein-Bialynicki approach. The chapter is closed with a discussion of response theory and Ein- stein causality.
On the Nonrelativistic Quantum-Mechanical Hamiltonian with 1/c21/c^21/c2 terms. Transverse current-current interaction
We extend the standard solid-state quantum mechanical Hamiltonian containing only Coulomb interactions between the charged particles by inclusion of 1/c21/c^21/c2 terms representing (transverse) current-current interaction. For its derivation we use the classical formulation of Landau-Lifshitz, however consequently in the Coulomb gauge. Our Hamiltonian does not coincide with the Darwin Hamiltonian and we emphasize the mathematical inconsistency in its derivation. We show, that the quantized version of our Hamiltonian is equivalent to the non-relativistic QED considering only states without photons and retaining only terms of order 1/c21/c^21/c2. The importance of this extended Hamiltonian lies in the possibility to distinguish external from internal magnetic fields. This aspect may be relevant for theories of the Meissner effect.
Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.
A field-quantized electromagnetic propagator description emphasizing the spatial localization of light emitted from a single-electron atom is presented. With the aim of obtaining a better insight into the quantum electrodynamics in the near-field zone of the atom, two approaches, formally different but equivalent in respect to their physical predictions, are studied. From the Maxwell equations, among the transverse operators, a transverse propagator picture in which the source region for the field is identified with the spatial domain occupied by the transverse current density of the atom emerges. In this picture the field propagation is isotropic and always retarded. By identifying the source region with that of the total atomic current, the transverse propagator becomes anisotropic, and attains a nonretarded self-field part and a near-field part different from zero only for spacelike events. By changing the Coulomb Lagrangian, a nonrelativistic Hamiltonian formalism adequate for reconciling the photon concept with the anisotropic propagator picture is introduced. In this formalism the transverse self-field energy is transferred to the particle Hamiltonian, and the interaction Hamiltonian now includes an interaction between the retarded field and the transverse self-field dynamics. The link between the standard (nonpropagator) theory, often used to argue that the total electromagnetic field is always retarded, and the transverse propagator theory is established. The equivalence between the two approaches is proved by demonstrating that the standard theory in fact includes a nonretarded response in the spatial domain occupied by the longitudinal part of the induced atomic current density.