An "Accidental" Symmetry Operator for the Dirac Equation in the Coulomb Potential

On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is intimately connected to the familiar Laplace-Runge-Lenz vector. Our approach guarantees not only derivation of Johnson-Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian follows.Comment: 6 page

Design of an electron microscope phase plate using a focused continuous-wave laser

We propose a Zernike phase contrast electron microscope that uses an intense laser focus to convert a phase image into a visible image. We present the relativistic quantum theory of the phase shift caused by the laser-electron-interaction, study resonant cavities for enhancing the laser intensity, and discuss applications in biology, soft materials science, and atomic and molecular physics.Comment: 5 pages, 3 figure