  The Gamma conjecture II for the quantum cohomology of a Fano manifold $F$,
proposed by Galkin, Golyshev and Iritani, describes the asymptotic behavior of
the flat sections of the Dubrovin connection near the irregular singularities,
in terms of a full exceptional collection, if there exists, of $\mathcal
D^b(F)$ and the $\widehat{\Gamma}$-integral structure. In this paper, for the
smooth quadric hypersurfaces we prove the convergence of the full quantum
cohomology and the Gamma conjecture II.
  For the proof, we first give a criterion on Gamma II for Fano manifolds with
semisimple quantum cohomology, by Dubrovin's theorem of analytic continuations
of semisimple Frobenius manifolds. Then we work out a closed formula of the
Chern characters of spinor bundles on quadrics. By the deformation-invariance
of Gromov-Witten invariants we show that the full quantum cohomology can be
reconstructed by its ambient part, and use this to obtain estimations. Finally
we complete the proof of Gamma II for quadrics by explicit asymptotic
expansions of flat sections corresponding to Kapranov's exceptional collections
and an application of our criterion.
