  We reexamine the relations between the Bethe-Salpeter (BS) wave function of
two particles, the on-shell scattering amplitude, and the effective potential
in quantum filed theory. It is emphasized that there is an exact relation
between the BS wave function inside the interaction range and the scattering
amplitude, and the reduced BS wave function, which is defined in this article,
plays an essential role in this relation. Based on the exact relation, we show
that the solution of Schr\"odinger equation with the effective potential gives
us a correct on-shell scattering amplitude only at the momentum where the
effective potential is calculated, while wrong results are obtained from the
Schr\"odinger equation at general momenta. We also discuss about a momentum
expansion of the reduced BS wave function and an uncertainty of the scattering
amplitude stemming from the choice of the interpolating operator in the BS wave
function. The theoretical conclusion obtained in this article could give hints
to understand the inconsistency observed in lattice QCD calculation of the
two-nucleon channels with different approaches.
