  Liouville transformations of Schr\"odinger equations preserve the scattering
amplitudes while changing the effective potential. We discuss the properties of
these gauge transformations and introduce a special Liouville gauge which
allows one to map the problem of quantum reflection of an atom on an attractive
Casimir-Polder well into that of reflection on a repulsive wall. We deduce a
quantitative evaluation of quantum reflection probabilities in terms of the
universal probability which corresponds to the solution of the $V_4=-C_4/z^4$
far-end Casimir-Polder potential.

Quantum Physics