Dirac-Coulomb Equation Research Articles

Exact solution of the N-dimensional generalized Dirac-Coulomb equation

An exact solution to the bound state problem for the N-dimensional generalized Dirac-Coulomb equation, whose potential contains both the Lorentz-vector and Lorentz-scalar terms of the Coulomb form, is obtained.

Bound state solutions of the two-electron Dirac-Coulomb equation

We show that the Dirac-Coulomb Hamiltonian hasnormalizable eigenfunctions corresponding to bound states.

Exact solution of the n-dimensional Dirac–Coulomb equation

An exact solution of the n-dimensional Dirac–Coulomb equation is obtained with the radial wave function containing only one term of a confluent hypergeometric function. It is of the same form as the solutions to the Schrödinger and Klein–Gordon equations with a Coulomb potential in n dimensions.

Correlation and relativistic effects for many-electron systems. II. Second-order energies for closed-shell atoms

The perturbation approach to the Dirac-Coulomb equation recently developed by the present authors is applied to the calculation of the energy correction at the first- and second-order level for a zero-order Hamiltonian taken as a symmetric sum of Fock operators. Results for several He-, Be- and Ne-like systems (up to Z=80) as well as for Zn2+ are presented and discussed.

Bound state solutions of the two-electron Dirac-Coulomb equation

Bound state solutions of the two-electron Dirac-Coulomb equation

Correlation and relativistic effects for many-electron systems

The N-electron Dirac-Coulomb equation is partitioned into an equivalent of the Schrödinger equation and perturbations. The perturbations involved are ordered according to increasing order of smallness in c−2. The electron-electron interaction itself is also treated as a perturbation. This leads to a new version of the double perturbation theory which provides the basic ideas for simultaneous study of the correlation and relativistic effects. Second-order results for some members of the He isoelectronic sequence are obtained by starting with the Hartree-Fock approximation and using the variational-perturbation method. The total energies calculated through second-order are in excellent agreement with their MCDF counterparts.

Relativistic perturbation theory. III. A new perturbation approach to the two-electron Dirac-Coulomb equation

For pt.II see ibid., vol.19, p.3431 (1986). Using the methods described by A. Rutkowski in pt.I (ibid., vol.19, p.149, 1986). A new perturbation approach is extended to two-electron problems. In this approach the Dirac-Coulomb equation is partitioned into an equivalent of the Schrodinger equation and a perturbation. The two perturbations involved each include a different negative power of the velocity of light. The relevant perturbation equations and formulae for the corrections to energy are derived. The method preserves some of the advantages of the Rayleigh-Schrodinger perturbation theory. A relativistic version of the variational perturbation method is formulated. In order to demonstrate the correctness of our two-electron relativistic perturbation theory, particular attention has been given to the exposition that the two-electron approach is consistent with the one-electron approach. It is shown that for the exact non-relativistic wavefunction the author's first-order energy correction is equal to the sum of appropriate quasirelativistic corrections. Numerical results are presented for the ground-state of helium-like atoms.

Infeld-Hull factorization and the simplified solution of the Dirac-Coulomb equation.

It is pointed out that the approach used by Su to obtain the simplified solution to the Dirac-Coulomb equation can be interpreted in the light of the Infeld-Hull factorization method. According to this method the first-order Dirac-Coulomb equation, after the transformation S-italic, necessarily implies two second-order ''Schroedinger-type'' equations, whose solutions are the upper and lower simplified solutions of the Dirac-Coulomb equation. It is also interesting to note that the ''recurrence relation'' between the upper and lower solutions, first obtained by Biedenharn, is exactly the Dirac-Coulomb equation itself, after the transformation S-italic.

Simplified solutions of the Dirac-Coulomb equation.

With the use of a simple similarity transformation which brings the radial wave equations of the Dirac-Coulomb problem into a form nearly identical to those of the Schr\"odinger and Klein-Gordon equations, we derive simplified solutions to the Dirac-Coulomb equation for both the bound and continuum states following the familiar standard procedure adopted in the derivation of the conventional solutions. We show that to obtain the desired form of the second-order radial equations we can still work with a first-order partial differential equation rather than with the second-order Dirac equation widely employed in the derivation of the simplified solutions, and thus we can avoid the task of reducing the solutions of the second-order equations to those of the original Dirac equation. The transformed Dirac-Coulomb radial equations are so simple that one can apply the WKB method to them in the same way that one applies the WKB approximation to the Schr\"odinger radial equation, without making the further approximations commonly invoked, and the Sommerfeld-Dirac discrete spectrum follows immediately. For small Z\ensuremath{\alpha}, we also present approximate expressions for both the transformed and the original Dirac-Coulomb wave functions, which are valid for all energies and certainly valid in the quasirelativistic approximation which takes account of relativistic effects. The structure of these approximate expressions also suggests one of the methods to obtain relativistic approximate wave functions from the Pauli-spinor wave function for more general central potentials arising in atoms, molecules, and solids.

Group theoretic treatment of the Dirac–Coulomb equation and matrix elements of its tensor operators

The solution of the Dirac–Coulomb equation obtained by Wong and Yeh is interpreted in terms of the SU(2)×SO(2,1) group. All electromagnetic transition probabilities can be considered as matrix elements of the tensor operators of this group, and evaluated exactly. The cases considered include transitions from bound-to-bound, bound-to-continuum, and continuum-to-continuum states.

Exact Path-Integral Solution of the Dirac-Coulomb Problem

An exact path-integral solution of the Dirac equation with a Coulomb potential is found. The energy-dependent Green's function of the second-order Dirac-Coulomb equation, expressed as a polar-coordinate path integral, is reduced to the exactly solvable path integral for an isotropic harmonic oscillator by a coordinate transformation combined with a local time rescaling. The Green's function of the linear Dirac equation is evaluated with the help of the Biedenharn transformation. The energy spectrum for the bound states is also obtained.

Exact solution of the Dirac-Coulomb equation and its application to bound-state problems. I External fields

An exact treatment of relativistic effects in bound-state problems in hydrogenlike atoms is given. In this paper we discuss the Zeeman effect and the Stark effect for relativistically bound electrons in a Coulomb field. For the Zeeman effect, the exact solution obtained by Darwin is used, and results of Crubellier and Feneuille for the Breit-Margenau correction are obtained in a simple way. For the Stark effect, the solution of Wong and Yeh is used. The exact expression can be readily compared with that of the Pauli approximation, and all correction terms can be identified. In the following paper we discuss interaction of the bound electron with radiation in an exact relativistic way. In the evaluation of the radial ${r}^{K}$ matrix elements with ${n}^{\ensuremath{'}}\ensuremath{\ne}n$, we present a closed-form expression as a sum over three parameters, derived from the method of the generating function of Laguerre polynomials.

Exact solution of the Dirac-Coulomb equation and its application to bound-state problems. II. Interaction with radiation

An exact treatment of relativistically bound electrons in hydrogenlike atoms interacting with radiation is given. The simplified solution of the Dirac-Coulomb equation obtained by Wong and Yeh is used in the calculation. Transition probabilities and sum rules are obtained. Our results can be readily compared with those of the Pauli approximation, and all correction terms can be identified. New sum rules have been obtained for the relativistic case which can be compared side by side with the corresponding ones in the nonrelativistic case.

Coulomb scattering of fast electrons

In a previous paper, a simple solution to the Dirac Coulomb equation has been obtained containing only one term of a confluent hypergeometric function for each component. This solution is now used to obtain an exact expression for the phase shift of relativistic electrons in Coulomb scattering, in complete accordance with the corresponding nonrelativistic case where the Schroedinger equation is used. Though the expression for the phase shift differs slightly from that of Mott, the resulting scattering cross section, to second order in ..cap alpha..Z and ..cap alpha..Z/..beta.., is the same as the one obtained correctly by McKinley and Feshbach. The asymmetry in double scattering is in agreement with Mott.

Simplified solution of the Dirac equation with a Coulomb potential

It is shown that the Dirac equation with a Coulomb potential has a simplified solution where each component contains one term of a confluent hypergeometric function only instead of two terms. This solution reduces to the usual free-field solution when the Coulomb potential is turned off. Thus the Dirac Coulomb equation has a solution which is not very different from the corresponding Schr\"odinger or Klein-Gordon equations.