  A previous result of the authors with Chaput and Perrin states that the union
of all rational curves of fixed degree passing through a Schubert variety in a
homogeneous space G/P is again a Schubert variety. In this paper we identify
this Schubert variety explicitly in terms of the Hecke product of Weyl group
elements. We apply our result to give an explicit formula for any two-point
Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula
and its equivariant generalization. We also recover a formula for the minimal
degree of a rational curve between two given points in a cominuscule variety.
