Mass dependence of Schroedinger wavefunctions

It is shown that for nonrelativistic central potentials of specific forms, the bound state wavefunctions u (r) have the property that G (r) equivalent∫ /sup r/0dx u (x)(partialu (x)/partialμ) > or =0 for all r. Here μ is the reduced mass. Thus, for such potentials, the probability that a particle lies within a spherical shell of radius r is a monotonically increasing function of μ. The forms for which this property has been established are (a) V (r) =Cr/sup epsilon/, -20 for nodeless states. It is shown explicitly that G (r) cannot be nonnegative for all bound states in an arbitrary monotonically increasing potential. The question remains open regarding the widest class of potentials for which G (r) is nonnegative for all bound states