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arXiv:1310.7567

Derivation of a four-spinor Dirac-like equation for the helium atom and an approximate ground state solution

D. L. Nascimento, A. L. A. Fonseca

Published Oct 26, 2013• 1 citation
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Abstract

We present a new analysis of the connection between the classical conservation theorems and the role played by the Dirac matrices in order to obtain a four spinor version of the Dirac equation for the two electrons bound problem. The separation of the set of equations in its angular and radial parts is done and an asymptotic solution for the radial system of equations is discussed for the ground state of the atom.

DOI
10.48550/arxiv.1310.7567

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We present a new analysis of the connection between the classical conservation theorems and the role played by the Dirac matrices in order to obtain a four spinor version of the Dirac equation for the two electrons bound problem. The separation of the set of equations in its angular and radial parts is done and an asymptotic solution for the radial system of equations is discussed for the ground state of the atom.
We present a two-dimensional analysis of the two-electron problem which comes from the classical conservation theorems and from which we obtain a version of the Dirac equation for the helium atom. Approximate solutions for this equation are discussed in two different methods, although in principle it can be solved analytically. One method is variational, of the Hylleraas type, the execution of which is left for a later communication. In contrast, the other method will have a more complete treatment, in which the set of equations will be separated into its angular and radial components. Furthermore, an exact solution for the angular component will be displayed as well as an approximate solution for the radial component, valid only for the fundamental state of the atom.
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.
The solution of the Dirac equation with the Coulomb potential is used to analyze bound electron states in a hydrogen atom. The analysis is based on the fact that such states are characterized by a set of quantum numbers which describe definite values of the complete set of physical quantities that can be determined simultaneously. This set includes the energy, square of the total angular momentum, one of its component and a spinor invariant. The latter, spinor invariant gives two-valued quantum number which determines the sign of its eigenvalue. In addition to the known Dirac and Johnson–Lippman invariant, there exists a new one. Operators of these three spinor invariants do not commute between themselves which results in the degeneracy of the energy levels with respect to the two-valued quantum number. Three different systems of the eigenbispinor corresponding to the three spinor invariants are obtained and the generalized solution with free parameters is calculated. Variation of the free parameters transforms one particular solution into any other. It is shown that the electron probability densities and spin polarizations in an electron cloud depend essentially on the invariant set, demonstrating physical difference of the states corresponding to different spinor invariants.
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.
The functional form of the wave function with respect to variables appropriate for the WKB approximation is chosen by taking into account the transformation properties of the spinor wave-function.This representation makes it possible, using only the wave equation, to pursue directly the time changes in the variables, i.e. the spin direction, velocity and density, etc., characterizing an electron.These changes correspond straightforwardly to a classical model of the non-relativistic Pauli electron with the proper magnetic moment, and the covariant form of this model, which is just "the Frenkel-Kramers model of the Dirac electron" introduced recently by the author.In this derivation, the existence of freedom of • rotation around the .spinaxis and certain definite angular velocity of this rotation are required.These results and the correspondence obtained may clarify the classical-mechanical significance of the spinor wave-function and also of the spin itself.An explanation is appended of the hyperbolic functions of the velocity metric, which help in the complicated calculations for the Dirac electron. § I. Introduction*) The main part of this paper appeared previously in Rep. Seisanken.No. 22 (1957), 55, in Japanese.
The WKB method is applied to solve the Dirac equation and the modified Dirac equation appropriate to a spin-\textonehalf{} particle with an anomalous magnetic moment. The solution consists of a phase factor multiplied by a spinor amplitude which is a power series in Planck's constant. The phase is a solution of the Hamilton-Jacobi equation of relativistic mechanics for a spinless particle without electric or magnetic moments. Each term in the spinor amplitude satisfies an ordinary differential equation along the relativistic trajectories. The equation for the leading amplitude yields an equation for the polarization four-vector which is identical with that derived classically by Bargmann, Michel, and Telegdi. It also yields the law of conservation of probability in a tube of trajectories. In addition, it gives rise to an equation for a supplementary phase factor. By using the classical Hamilton-Jacobi function, the law of probability conservation, the polarization four-vector and the supplementary phase factor, the leading term in the solution of the Dirac or modified Dirac equation can be constructed. This solution should be useful when the wavelength of the particle is small compared to the characteristic distance associated with the electromagnetic potential through which the particle moves. When applied to the bound states of a particle without an anomalous moment in a spherically symmetric electrostatic potential, it yields the same results as are usually obtained by separation of variables and use of the ordinary WKB method. The advantage of the present method is that it applies equally well to nonseparable problems.