Issue with Fisher Information and the Correspondence Principle?

The classical Fisher information, F, in one dimension is given in the literature as:

Integral dx d(x) [ d/dx ln( d(x) ) ]2  ((1)) where d(x) is the spatial probability density. For quantum mechanics, (1) uses d(x)=W*(x)W(x) for quantum mechanics, where W(x) is the complex wavefunction. The Cramer Rao inequality is used to show how 1/ F is related to the variance of d(x). For a bound state wavefunction, which is real, one may show Fisher’s information leads to Integral dx v(x)v(x) d(x) which is proportional to the average kinetic energy. Using the correspondence principle, d(x), for high energy levels, should approach C/v(x) where v(x) is the classical velocity. On the other hand, using the definition of F directly with d(x)=C/v(x) leads to a different result. This seems unusual. We try to investigate in this note to see that quantum features exist even in the high energy limit. The Fisher information calculated with W(x)W(x)=C/v(x) seems to capture quantum information, while that with d(x)=C/v(x), information about the envelope function.