  We proposed a formally exact, probabilistic method to assess the validity of
the Thomas-Fermi potential for three-dimensional condensed matter systems where
electron dynamics is constrained to the Fermi surface. Our method, which relies
on accurate solutions of the radial Schr\"{o}dinger equation, yields the
probability density function for momentum transfer. This allows for the
computation of its expectation values, which can be compared with unity to
confirm the validity of the Thomas-Fermi approximation. We applied this method
to three {\it n}-type direct-gap III-V model semiconductors (GaAs, InAs, InSb)
and found that the Thomas-Fermi approximation is certainly valid at high
electron densities. In these cases, the probability density function exhibits
the same profile, irrespective of the material under scrutiny. Furthermore, we
show that this approximation can lead to serious errors in the computation of
observables when applied to GaAs at zero temperature for most electron
densities under scrutiny.
