On the Algebraic Derivation of Phase Shifts for Shape Invariant Potentials

Abstract

Phase shifts produced by two supersymmetric partner potentials satisfy a simple relation. One expects that the phase shift can be obtained algebraically, if the potential is shape invariant. We show that this is true only for a special class of shape invariant potentials. Among known shape invariant potentials, only the Coulomb potential satisfy this requirement. The Coulomb phase shift can be obtained analytically, which agrees with the well-known standard result.

Phase Shift between Supersymmetric Partner Potentials

“…Such a restrictive condition is valid only for the Coulomb potential [4], only for which analytic expression for the phase shift is possible [5][6][7].…”

“…Hence in this case, use of Equation (20) permits the derivation of an algebraic expression for the phase shift. This is true only for the Coulomb potential [5,7]. In the above derivation, we have not assumed the shape invariance property [4].…”

Phase shift of isospectral potentials

“…Moreover, as the original potential and the isospectral potential produce same phase shift corresponding to a particular energy and l value, the phase shift relation [Eq. (2)] is not valid for isospectral potentials whereas it is valid for supersymmetric partner potentials [6,7].…”

“…where k is the wave number of the incident wave and γ 0 = 2mB0 2 , B 0 being the binding energy of the ground state in V (1) . An exhaustive study [6,7] has been done to examine the validity of Eq. (2) for Supersymmetric partner potentials.…”

“…It is also shown that the phase shifts of shape invariant potentials can be derived analytically only if the potential together with the centrifugal repulsion corresponding to the orbital angular momentum l is shape invariant and its partner involves centrifugal repulsion corresponding to orbital angular momentum (l + 1). Among the known SIPs related by translation of parameters [4], which have continuum spectra, only the Coulomb potential satisfies this criterion [6]. Except the Coulomb potential, no other SIP has the orbital angular momentum l as the shape invariance parameter.…”