Fermi-Segrè Formula

For a nonrelativistic quantal particle moving with the angular momentum $l=0\mathrm{}$ in a spherically symmetric, smooth potential well, which in the neighborhood of the center of force gives rise to an attractive Coulomb force, a formula for the probability density of the particle at the center of force is derived by means of certain phase-integral approximations of an arbitrary order. From this formula, which is expected to be very accurate, the non-relativistic approximation of the Fermi-Segrè formula for the probability density of finding an $s$ electron at the nucleus of an atom or ion is obtained. The limits of validity of the Fermi-Segrè formula in the nonrelativistic approximation are clarified by this derivation.