Quantum cohomology of projective bundles over $\mathbb{P}^{n}$

In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space $\mathbb{P}^{n}$. Using excessive intersection theory, we compute the leading coefficients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev's conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over $\mathbb{P}^{n}$ is partially verified. Moreover, relations between the quantum cohomology ring structure and Mori's theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.