GENERALIZATION OF THE FERMI-SEGRE FORMULA

A generalization of the non-relativistic Fermi-Segre formula into a formula which is valid also for angular momentum quantum numbers l different from zero, is derived by means of a phase-integral method. The formula thus obtained, which gives an expression for the limit of u(r)/r**l** plus **1 as r yields 0, where u(r) is a normalized bound-state radial wavefunction, in terms of the derivative of the energy level E//n// prime with respect to the radial quantum number n+40,is an improvement and generalization of a formula which has been obtained by M. A. Bouchiat and C. Bouchiat. It reduces to their formula for a particular class of potentials and highly excited states with not too large values of l, and it reduces to the Fermi-Segre formula when l equals 0. The accuracy of the authors' formula, as well as that of the Bouchiat-Bouchiat formula, is investigated by application to an exactly soluble model. The formula obtained can also be written in another form by replacing dE//n// prime /dn prime by an expression involving a closed-loop integral in the complex r-plane (around the generalized classical turning points), the integrand being a phase-integral quantity expressed in terms of the potential in which the particle moves. It is also shown that the exact value of the limit of u(r)/r**l** plus **1 as r yields 0 can be expressed as an expectation value of a certain function depending on the physical potential V(r) and r as well as on l and E//n// prime .