Upper and lower bounds on the mean square radius and criteria for the occurrence of quantum halo states

Abstract

In the context of nonrelativistic quantum mechanics, we obtain several upper and lower limits on the mean square radius applicable to systems composed of two bodies bound by a central potential. A lower limit on the mean square radius is used to obtain a simple criterion for the occurrence of S-wave quantum halo sates.

Transition states and the critical parameters of central potentials

“…Substituting representation (18) into equation (9), and then equating the expansion coefficients of the same powers of r for the left-hand (lhs) and right-hand sides (rhs) of equation ( 9), one can calculate any finite number of the subsequent coefficients c i and b i with i 1.…”

“…The straightforward solution of the second-order differential equation (9) with the boundary conditions (17) and (24) presents our first method for calculating the critical parameters β n of a given attractive potential (3) satisfying the boundary conditions (1) and (2). This method is especially effective and accurate for small values of l. For a few potentials, such as the exponential, the Hulthen and the Woods-Saxon, one can derive analytical expressions for the critical parameters using this method (see the appendix).…”

“…This method is especially effective and accurate for small values of l. For a few potentials, such as the exponential, the Hulthen and the Woods-Saxon, one can derive analytical expressions for the critical parameters using this method (see the appendix). However, this is possible only for S-states (l = 0), when equation (9) with the potentials mentioned above has a general analytical solution. We are familiar with only one form of central potential which admits an analytical solution of equation ( 9) for l 0.…”

“…For s = 0, one obtains the potential which is widely known as the finite square well. Equation (9) with the potential (A.1) reduces to the form…”

“…Since then many authors have contributed to the study of this problem. Among them we would like to refer to the works [3][4][5][6][7] and to emphasize particularly the contribution of Calogero and Brau [8][9][10][11][12].…”