Second-Order Stationary Solutions for Fermions in an External Coulomb Field

Abstract

We have studied self-conjugate second-order equations with spinor wavefunctions for fermions moving in an external Coulomb field. For stationary states, the equations are characterized by separated states with positive and negative energies, which render probabilistic interpretation possible. For the Coulomb field of attraction, the energy spectrum of the second-order equation coincides with the spectrum of the Dirac equation, while the probability densities of states are slightly different. For a Coulomb field of repulsion, there exists an impermeable potential barrier with radius depending on the classical electron radius and on the electron energy. The existence of the impermeable barrier does not contradict the results of experiment for determining the inner electron structure and does not affect (in the lowest order of perturbation theory) the Coulomb electron scattering cross section. The existence of the impermeable barrier can lead to positron confinement in supercritical nuclei with Z ≥ 170 in case of realization of spontaneous emission of vacuum electron–positron pairs.

Pruefer Transformation and Boundary Conditions

For numerical solution of Eqs. (39) and (40), we have used the Pruefer transformation [20–23].

By way of example, let us consider Eq. (39).

For function f(r) = rF(r), Eq. (39) has form

$$\frac{{{{d}^{2}}f(r)}}{{d{{r}^{2}}}} + 2({{E}_{{{\text{Schr}}}}} - U_{{{\text{eff}}}}^{2}(r))f(r) = 0.$$

((A.1))

Let us assume that

$$\begin{gathered} f(r) = P(r)\sin \Phi (r), \\ \frac{{df(r)}}{{dr}} = P(r)\cos \Phi (r). \\ \end{gathered} $$

((A.2))

Then

$$\frac{{f(r)}}{{df(r){\text{/}}dr}} = \tan \Phi (r)$$

((A.3))

and Eq. (A.1) can be written in the form of the following system of first-order differential equations:

$$\frac{{d\Phi }}{{dr}} = {{\cos }^{2}}\Phi + 2({{E}_{{{\text{Schr}}}}} - U_{{{\text{eff}}}}^{F}){{\sin }^{2}}\Phi ,$$

((A.4))

$$\frac{{d\ln P}}{{dr}} = (1 - 2({{E}_{{{\text{Schr}}}}} - U_{{{\text{eff}}}}^{F}))\sin \Phi \cos \Phi .$$

((A.5))

It should be noted that Eq. (A.5) must be solved after determining eigenvalues ε_n and eigenfunctions Φ_n(r) from Eq. (A.4)

For bound state with r → ∞, we get

$$\begin{gathered} \tan \Phi {{{\text{|}}}_{{r \to \infty }}} = - \frac{1}{{\sqrt {1 - {{\varepsilon }^{2}}} }}, \\ \Phi {{{\text{|}}}_{{r \to \infty }}} = - \arctan \frac{1}{{\sqrt {1 - {{\varepsilon }^{2}}} }} + k\pi . \\ \end{gathered} $$

((A.6))

For exponentially increasing solutions for r → ∞, we have

$$\begin{gathered} \tan \Phi {{{\text{|}}}_{{r \to \infty }}} = \frac{1}{{\sqrt {1 - {{\varepsilon }^{2}}} }}, \\ \Phi {{{\text{|}}}_{{r \to \infty }}} = \arctan \frac{1}{{\sqrt {1 - {{\varepsilon }^{2}}} }} + k\pi . \\ \end{gathered} $$

((A.7))

In expressions (A.6) and (A.7), k = 0, ±1, ±2, … .

The numerical method for solving equations of type (A.4), (A.5) with asymptotic forms (A.6) and (A.7) is described in detail in [3–5, 27].