WKB method for the Dirac equation with central-symmetrical potential and its application to the theory of two dimensional supercritical atoms

Abstract

Solutions of the Dirac equation in a strong external field are obtained in the WKB approximation. A field is considered strong if the electron binding energy exceeds 2mc2 and the discrete spectrum levels may be lowered into the lower continuum. The wave functions in the classically allowed and forbidden regions are found and the conditions for matching them on transition through the turning point are obtained. The WKB method is applied to the following problems: 1) generalization of the Bohr–Sommerfeld quantization conditions with allowance for relativistic effects and the spin in 2 + 1 dimensions; 2) energy and width of the quasistationary level in the lower continuum.