arXiv subjects

Robert J. Ducharme

Publications and source records attributed to Robert J. Ducharme.

14 recordsLinked to original sources

arXiv2020

Fractional Angular Momenta, Gouy and Berry phases in Relativistic Bateman-Hillion-Gaussian Beams of Electrons

A new Bateman-Hillion solution to the Dirac equation for a relativistic Gaussian electron beam taking explicit account of the $4$-position of the beam waist is presented. This solution has a pure Gaussian form in the paraxial limit but beyond it contains higher order Laguerre-Gaussian components attributable to the tighter focusing. One implication of the mixed mode nature of strongly diffracting beams is that the expectation values for spin and orbital angular momenta are fractional and are interrelated to each other by \textit{intrinsic spin-orbit coupling}. Our results for these properties align with earlier work on Bessel beams [Bliokh \textit{et al.} Phys. Rev. Lett. \textbf{107}, 174802 (2011)] and show that fractional angular momenta can be expressed by means of a Berry phase. The most significant difference arises, though, due to the fact that Laguerre-Gaussian beams naturally contain Gouy phase, while Bessel beams do not. We show that Gouy phase is also related to Berry phase and that Gouy phase fronts that are flat in the paraxial limit become curved beyond it.

quant-ph↗

arXiv2016

Lorentz Covariant Quantum 4-Potential and Orbital Angular Momentum for the Transverse Confinement of Matter Waves

In two recent papers exact Hermite-Gaussian solutions to relativistic wave equations have been obtained for both electromagnetic and particle beams that include Gouy phase. The solutions for particle beams correspond to those of the Schrödinger equation in the non-relativistic limit. Here, distinct canonical and kinetic 4-momentum operators will be defined for quantum particles in matter wave beams. The kinetic momentum is equal to the canonical momentum minus the fluctuating terms resulting from the transverse localization of the beam. Three results are obtained. First, the total energy of a particle for each beam mode is calculated. Second, the localization terms couple into the canonical 4-momentum of the beam particles as a Lorentz covariant quantum 4-potential originating at the waist. The quantum 4-potential plays an analogous role in relativistic Hamiltonian quantum mechanics to the Bohm potential in the non-relativistic quantum Hamilton-Jacobi equation. Third, the orbital angular momentum (OAM) operator must be defined in terms of canonical momentum operators. It is further shown that kinetic 4-momentum does not contribute to OAM indicating that OAM can therefore be regarded as a pure manifestation of quantum 4-potential.

quant-ph↗

arXiv2015

Gouy Phase for Relativistic Quantum Particles

Recently Gouy rotation was observed with focused non-relativistic electron vortex beams. If the electrons in vortex beams are very fast we have to take into account relativistic effects to completely describe the Gouy phase on them. Exact Hermite-Gaussian solutions to the Klein-Gordon equation for particle beams are obtained here that make explicit the 4-position of the focal point of the beam. These are Bateman-Hillion solutions with modified phase factors to take into account the rest mass of the particles. They enable a relativistic expression for the Gouy phase to be determined. It is in fact shown all the solutions are form invariant under Lorentz transformations. It is further shown for the exact solutions to correspond to those of the Schrödinger equation the relative time between the focal point and any point in the beam must be constrained to be a specific function of the relative spatial coordinates.

quant-ph↗

arXiv2014

Constrained Bateman-Hillion Solutions for Hermite-Gaussian Beams

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.

quant-ph↗

arXiv2014

A Unified Understanding of Spin and Orbital Angular Momentum in the Complex Plane

The quantum mechanical operator for angular momentum is transformed from the real plane into the complex plane. In doing so, the Cauchy-Riemann (C-R) equations are interpreted as constraint conditions defining two distinct domains where complex differentiation is permitted. It is shown each of these domains contains an orbital angular momentum contribution plus an non-orbital term that cancels out between them. It is further shown the field equations for spinning quantum particles include C-R equations that restrict the particles to a single complex constraint space. It is therefore proposed the non-orbital term in the constraint space angular momentum is the source of the spin.

physics.gen-ph↗

arXiv2012

A Solution Technique for Quantum Mechanical Differential Equations Using Multiple Complex Planes

It is shown fields that cannot be represented over one complex plane can be further decomposed for representation over multiple complex planes. This finding is demonstrated here by solving of the Schrödinger equation for the hydrogen atom in a complex space containing three complex planes. The complex coordinate system is generated from real coordinates using an isometric transformation. One plane is applied to mix energy and time; the other two planes are used to represent the z-component of angular momentum of the electron. The eigensolutions of the Schrödinger equation are shown to be holomorphic in the complex planes.

math-ph↗

arXiv2011

Conjugate Spinor Solution of the Dirac Equation for the Hydrogen Atom

It is shown the central field Dirac equation can be simplified through the use of real conjugate spinors to substitute for the upper and lower components of the bi-spinor eigensolutions. This substitution reduces the Dirac equation for the hydrogen atom to the problem of solving a single second order differential equation similar to a Klein-Gordon equation but containing additional terms to take account of the spin on the electron. The bi-spinor wave functions are readily constructed once the solution is known in terms of the conjugate spinors.

quant-ph↗

arXiv2011

Integral Transforms in Relativistic Quantum Constraint Mechanics

In relativistic quantum constraint mechanics the state of a physical system is constrained to a 3-dimensional subspace of Minkowski 4-space. Fourier transformation can be used to relate this state between constraint spaces in 4-position and 4-momentum space. It is shown that integral transforms of this nature can be carried out using Lorentz-invariant 3-dimensional constraint space coordinates such that a complete equivalence class of 4-space representations can be constructed from the transform. This method is further applied to develop a relativistic generalization of the Segal-Bargmann transformation that leads to the representation of quantum systems in a three-dimensional subspace of Bargmann 4-space.

quant-ph↗

arXiv2010

Isometric Entanglement of Particle Positions in Quantum Bound Systems

It is shown the role of a scalar potential in the Schrödinger equation for a steady-state two-particle system is equivalent to an isometric entanglement of the position coordinates of the particles in space and time. The entangled coordinates of each particle are complex quantities related through the entangling transformation to the real positions of both particles. The transformation takes into account all of the states in the Hilbert space of the composite system. Transforming the Schrödinger equation into these entangled coordinates eliminates the scalar potential.

quant-ph↗

arXiv2010

Exact Solution of the Klein-Gordon Equation for the Hydrogen Atom Including Electron Spin

The term describing the coupling between total angular momentum and energy-momentum in the hydrogen atom is isolated from the radial Dirac equation and used to replace the corresponding orbital angular momentum coupling term in the radial K-G equation. The resulting spin-corrected K-G equation is a second order differential equation that contains no matrices. It is solved here to generate the same energy eigenvalues for the hydrogen atom as the Dirac equation.

quant-ph↗

arXiv2010

Conformal Mapping of Relativistic Quantum Bound Systems to Eliminate Potential Fields

In two recent papers, an isometric conformal transformation has been introduced that eliminates potential interaction terms from the Schrödinger equation for central potential problems. The method has been demonstrated for both the hydrogen atom and three-dimensional harmonic oscillator. Here, it is shown that the same transformation technique can also be applied to central potential problems formulated using the Klein-Gordon equation.

quant-ph↗

arXiv2010

Conformal Transformation of the Schrödinger Equation for Central Potential Problems in Three-Dimensions

In a recent paper, it has been shown the Schrödinger equation for the three-dimensional harmonic oscillator can be simplified through the use of an isometric conformal transformation. Here, it is demonstrated that the same transformation technique is also applicable to the Schrödinger equation for the hydrogen atom. This approach has two interesting features. Firstly, it eliminates potential fields from the Schrödinger equation. The Coulomb and harmonic binding terms are instead represented as imaginary parts of complex time. Secondly, the method leads to a general relationship between potential energy and ground state energy that encompasses both the hydrogen atom and the harmonic oscillator as special cases.

quant-ph↗

arXiv2010

Conformal Transformation of the Schrödinger Equation for the Harmonic Oscillator into a Simpler Form

The Schrödinger equation and ladder operators for the harmonic oscillator are shown to simplify through the use of an isometric conformal transformation. These results are discussed in relation to the Bargmann representation. It is further demonstrated that harmonic interactions can be introduced into quantum mechanics as an imaginary component of time equivalent to adding the oscillator potential into the hamiltonian for the confined particle.

quant-ph↗

arXiv2009

Relativistic Ladder Operators for the Three-Dimensional Harmonic Oscillator

The quantum constraint equations for a relativistic three-dimensional harmonic oscillator are shown to find concise expression in terms of Lorentz covariant ladder operators. These ladder operators consist of two conjugate 4-vectors that are each constrained to generate three linearly independent combinations of ladder operator components for raising and lowering the eigenstates of the oscillator. Correspondence to the Schrödinger equation for the harmonic oscillator in the non-relativistic limit is demonstrated.

quant-ph↗