  Let V be a vector space with a nondegenerate symmetric form and OG be the
orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We
give a presentation for the (small) quantum cohomology ring QH^*(OG) and show
that its product structure is determined by the ring of (P~)-polynomials. A
"quantum Schubert calculus" is formulated, which includes quantum Pieri and
Giambelli formulas, as well as algorithms for computing Gromov-Witten
invariants. As an application, we show that the table of 3-point, genus zero
Gromov-Witten invariants for OG coincides with that for a corresponding
Lagrangian Grassmannian LG, up to an involution.
