General solution vs spin invariant eigenstates of the Dirac equation
with the Coulomb potential
- URL: http://arxiv.org/abs/2111.08552v1
- Date: Tue, 16 Nov 2021 15:27:56 GMT
- Title: General solution vs spin invariant eigenstates of the Dirac equation
with the Coulomb potential
- Authors: L.S. Brizhik, A.A. Eremko, V.M. Loktev
- Abstract summary: Solutions of the Dirac equation for an electron in the Coulomb potential are obtained using operator invariants of the equation.
It is shown for the first time that these invariants determine electron spatial probability amplitude and spin polarization in each quantum state.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Solutions of the Dirac equation for an electron in the Coulomb potential are
obtained using operator invariants of the equation, namely the Dirac,
Johnson-Lippmann and recently found new invariant. It is demonstrated that
these operators are the spin invariants. The generalized invariant is
constructed and the exact general solution of the Dirac equation are found. In
particular, the explicit expressions of the bispinors corresponding to the
three complete sets of the invariants, their eigenvalues and quantum numbers
are calculated. It is shown that the general solution of one center Coulomb
Dirac equation contains free parameters. Changing one or more of these
parameters, one can transform one solution of the Dirac equation into any
other. It is shown for the first time that these invariants determine electron
spatial probability amplitude and spin polarization in each quantum state.
Electron probability densities and spin polarizations are explicitly calculated
in the general form for several electron states in the hydrogen-like energy
spectrum. Spatial distributions of these characteristics are shown to depend
essentially on the invariant set, demonstrating, in spite of the accidental
degeneracy of energy levels, physical difference of the states corresponding to
different spin invariants.
Related papers
- Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators [7.694970944345054]
We provide a general formula in terms of $mathscrS_texto$ and $vecmathscrP_texto$ for the total charge of any subregion of the system.
Results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently.
arXiv Detail & Related papers (2024-10-04T18:00:01Z) - Relativistic exponential-type spinor orbitals and their use in many-electron Dirac equation solution [0.0]
Dirac-Coulomb type differential equation and its solution relativistic exponential-type spinor orbitals are introduced.
A new formulation for relativistic auxiliary functions that improve the efficiency in Coulomb energy calculations is presented.
arXiv Detail & Related papers (2024-03-23T20:48:54Z) - Algebra of the spinor invariants and the relativistic hydrogen atom [0.0]
It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation.
It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom.
arXiv Detail & Related papers (2022-11-03T14:50:01Z) - New symmetries, conserved quantities and gauge nature of a free Dirac
field [0.0]
We present and amplify some of our previous statements on non-canonical interrelations between the solutions to free Dirac equation (DE) and Klein-Gordon equation (KGE)
We demonstrate that all the solutions to the DE can be obtained via differentiation of a corresponding pair of the KGE solutions for a doublet of scalar fields.
arXiv Detail & Related papers (2022-08-31T13:49:49Z) - Relativistic dynamical inversion in manifestly covariant form [0.0]
The Relativistic Dynamical Inversion technique is a novel tool for finding analytical solutions to the Dirac equation.
The most remarkable feature of the new method is the ease of performing non-trivial change of reference frames.
A whole family of normalizable analytic solutions to the Dirac equation is constructed.
arXiv Detail & Related papers (2022-05-24T10:19:28Z) - Manipulating Generalized Dirac Cones In Quantum Metasurfaces [68.8204255655161]
We consider a collection of single quantum emitters arranged in a honeycomb lattice with subwavelength periodicity.
We show that introducing uniaxial anisotropy in the lattice results in modified dispersion relations.
arXiv Detail & Related papers (2022-03-21T17:59:58Z) - Traveling Wave Form Description for Dirac Field and Its Deduction To
Pauli Equation Type Forms in Quantum Mechanics [7.6915316507201785]
We derive an equivalent traveling wave form description for Dirac field.
In the non-relativistic limit, such form can reduce to inverse-Galilean transformed Schrodinger-type equation.
arXiv Detail & Related papers (2022-03-17T00:48:50Z) - Out-of-equilibrium dynamics of the Kitaev model on the Bethe lattice via
coupled Heisenberg equations [23.87373187143897]
We study the isotropic Kitaev spin-$1/2$ model on the Bethe lattice.
We take a straightforward approach of solving Heisenberg equations for a tailored subset of spin operators.
As an example, we calculate the time-dependent expectation value of this observable for a factorized translation-invariant.
arXiv Detail & Related papers (2021-10-25T17:37:33Z) - Deformed Explicitly Correlated Gaussians [58.720142291102135]
Deformed correlated Gaussian basis functions are introduced and their matrix elements are calculated.
These basis functions can be used to solve problems with nonspherical potentials.
arXiv Detail & Related papers (2021-08-10T18:23:06Z) - General solution of the Dirac equation with the Coulomb potential [0.0]
The general solution of the Dirac equation with the Coulomb potential is shown to contain free parameters.
The spatial distributions of these characteristics are shown to depend essentially on the invariant set.
arXiv Detail & Related papers (2020-09-17T10:39:03Z) - General quantum-mechanical solution for twisted electrons in a uniform
magnetic field [68.8204255655161]
A theory of twisted (and other structured) paraxial electrons in a uniform magnetic field is developed.
The observable effect of a different behavior of relativistic Laguerre-Gauss beams with opposite directions of the orbital angular momentum penetrating from the free space into a magnetic field is predicted.
arXiv Detail & Related papers (2020-05-13T16:35:10Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.