Abstract
In several perturbative calculations, such as the Lamb shift and decay of positronium into two photons, it is necessary to calculate the wave function of the S state, or the first derivative of the P state, etc. at the origin. This value diverges for the Dirac-Coulomb wave function. A method suggested by Bethe and Salpeter is used and the result is obtained in analytically closed form. For the hydrogen atom in the ground state (1${S}_{1/2}$) our result for ${u}^{2}$(0) is equal to the traditional result, calculated by using the nonrelativistic Schr\"odinger wave function, multiplied by a factor slightly greater than unity, 1.000 118. For the 2${S}_{1/2}$ state, the multiplication factor is 1.000 004 1.
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