APPROXIMATE TREATMENT OF THE DIRAC-EQUATION WITH SCALAR AND VECTOR POTENTIALS OF RECTANGULAR SHAPES

The Dirac equation with a scalar potential U(S)(r) and the fourth comp onent of a vector potential U(V)(r) is considered in the case of recta ngular shapes of these potentials with the same radius R; approximate analytic expressions are derived for the single-particle energy of bou nd states in certain cases. The results obtained with these expression s are compared with the corresponding ''exact'' results obtained by so lving the eigenvalue equation numerically. It is found that very good results are obtained for the ground state and for quite a wide range o f values of R with one of the proposed expressions. Even the correspon ding nonrelativistic version of this expression has not been derived b efore, to our knowledge.