Let V be a vector space with a non-degenerate 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 three-point, genus zero Gromov–Witten invariants for OG coincides with that for
a corresponding Lagrangian Grassmannian LG, up to an involution.