Lorentz Covariant Quantum 4-Potential and Orbital Angular Momentum for the Transverse Confinement of Matter Waves
arXiv:1601.07544 · doi:10.1103/PhysRevA.94.023822
Abstract
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.
9 pages
References in corpus (9)
- Semiclassical Dynamics of Electron Wave Packet States with Phase Vortices
- Electron vortex beams in a magnetic field: A new twist on Landau levels and Aharonov-Bohm states
- Atomic scale electron vortices for nanoresearch
- Colliding particles carrying non-zero orbital angular momentum
- Imaging the dynamics of free-electron Landau states
- Cosmology from quantum potential
- A Raman Waveplate for Spinor BECs
- Relativistic electron vortex beams in a laser field
- Gouy Phase for Relativistic Quantum Particles