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

arXiv:2307.04826 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Paraxial Dirac equation

Tomasz Radożycki

Published Dec 1, 2023• 0 citations
arXivHTMLPDF

Abstract

In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac equation is derived, and then its solutions in the aforementioned cases are directly obtained. All the resulting wave functions conform to those derived formerly by virtue of superpositions, except for terms that are negligible upon the assumption that the paraxial functions along the propagation axis vary only slightly over a distance equal to the de Broglie wavelength, which is the standard paraxial requirement.

DOI
10.1098/rspa.2023.0493

Related papers

Paraxial quantum optics

J. C. Garrison, R. Y. Chiao

The classical concept of a paraxial wave is extended to quantum optics as a condition on the state vector, and the family of vectors obeying this condition defines a paraxial Hilbert space. Field operators acting on this space are approximated by the product of a slowly-varying envelope operator and a classical carrier wave. This permits the discussion of Gaussian pulses and paraxial wave packets. This treatment is extended to obtain an expansion of state vectors and field operators in powers of the opening angle of the cone of paraxial rays. The leading term in the expansion yields the component of the angular momentum along the propagation axis as a sum of well defined orbital- and spin-parts. With the same technique, it is shown that approximate localization of photons is possible for localization volumes much larger that the mean wavelength in the paraxial state.

Oxford University Press eBooks•Jun 5, 2008•0 citations

Exact quantum-mechanical equations for particle beams

Alexander J. Silenko

The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.

arXiv:2206.14901•Modern Physics Letters A•May 30, 2022•8 citations

Dirac equation for electrodynamic model particles

J. X. Zheng-Johansson

We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.

Journal of Physics Conference Series•Aug 1, 2008•5 citations

The paraxial approximation fails to describe the interaction of atoms with general vortex light fields

G. F. Quinteiro, Christian Tomás Schmiegelow, Ferdinand Schmidt‐Kaler

Neglecting the electric field component along the light's propagation direction is a common practice, known as paraxial approximation. However, experimental evidence and theory on the head-on excitation of atoms by Laguerre-Gaussian beams reveal that the full vector character of the light field has to be taken into account. Optical vortices are a large family of fields, being Laguerre-Gaussian only one particular case. Now we extend the study of the applicability of the paraxial approximation to a broader set of optical vortices by considering the interaction of atoms with generalized Laguerre-Gaussian and generalized Bessel beams. We demonstrate that all vortex beams here considered look the same close to the beam axis, where the atom is placed. From this, we conclude that the paraxial approximation fails for the large set of vortex beams, and that a full understanding of the interaction of optical vortices with atoms requires the inclusion of all vector components of the electric field.

arXiv:2004.00040•Mar 31, 2020•3 citations

Quantization of Hermite–Gaussian and Laguerre–Gaussian beams and their spatial transformations

Vlasta Peřinová, A. Lukš

The paraxial quantum propagation of wavepackets is studied. Hermite–Gaussian and Laguerre–Gaussian beams are considered with respective expansions. A description of free-space propagation and transformation of radiation by optical system is worked out.

Journal of Modern Optics•Feb 21, 2006•11 citations

Constrained Bateman-Hillion Solutions for Hermite-Gaussian Beams

Ducharme, Robert J.

Exact Bateman-Hillion solutions of the wave equation are applied to Hermite-Gaussian beams using a space-time constraint condition that requires the field density to fall as the inverse square of distance from the focal point of the beam at large distances from it. Following a familiar practice, the constraint is implemented in integrals through the use of a Dirac delta function. It is shown the Hermite-Gaussian functions evolve to become pure functions of angular position on the fully developed spherical phase fronts. Under the paraxial approximation it is further shown the wave equation and Schrodinger equation are interchangeable within the constraint space in correspondence to a recent paper claiming indirect evidence of Gouy phase in matter waves.

arXiv:1412.1928•Dec 5, 2014•1 citation
In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac equation is derived, and then its solutions in the aforementioned cases are directly obtained. All the resulting wave functions conform to those derived formerly by virtue of superpositions, except for terms that are negligible upon the assumption that the paraxial functions along the propagation axis vary only slightly over a distance equal to the de Broglie wavelength, which is the standard paraxial requirement.
The classical concept of a paraxial wave is extended to quantum optics as a condition on the state vector, and the family of vectors obeying this condition defines a paraxial Hilbert space. Field operators acting on this space are approximated by the product of a slowly-varying envelope operator and a classical carrier wave. This permits the discussion of Gaussian pulses and paraxial wave packets. This treatment is extended to obtain an expansion of state vectors and field operators in powers of the opening angle of the cone of paraxial rays. The leading term in the expansion yields the component of the angular momentum along the propagation axis as a sum of well defined orbital- and spin-parts. With the same technique, it is shown that approximate localization of photons is possible for localization volumes much larger that the mean wavelength in the paraxial state.
The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.
We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.
Neglecting the electric field component along the light's propagation direction is a common practice, known as paraxial approximation. However, experimental evidence and theory on the head-on excitation of atoms by Laguerre-Gaussian beams reveal that the full vector character of the light field has to be taken into account. Optical vortices are a large family of fields, being Laguerre-Gaussian only one particular case. Now we extend the study of the applicability of the paraxial approximation to a broader set of optical vortices by considering the interaction of atoms with generalized Laguerre-Gaussian and generalized Bessel beams. We demonstrate that all vortex beams here considered look the same close to the beam axis, where the atom is placed. From this, we conclude that the paraxial approximation fails for the large set of vortex beams, and that a full understanding of the interaction of optical vortices with atoms requires the inclusion of all vector components of the electric field.
Exact Bateman-Hillion solutions of the wave equation are applied to Hermite-Gaussian beams using a space-time constraint condition that requires the field density to fall as the inverse square of distance from the focal point of the beam at large distances from it. Following a familiar practice, the constraint is implemented in integrals through the use of a Dirac delta function. It is shown the Hermite-Gaussian functions evolve to become pure functions of angular position on the fully developed spherical phase fronts. Under the paraxial approximation it is further shown the wave equation and Schrodinger equation are interchangeable within the constraint space in correspondence to a recent paper claiming indirect evidence of Gouy phase in matter waves.