There exists a remarkable connection between the quantum mechanical
Landau-Zener problem and purely classical-mechanical problem of a ball rolling
on a Cornu spiral. This correspondence allows us to calculate a complicated
multiple integral, a kind of multi-dimensional generalization of Fresnel
integrals. A direct method of calculation is also considered but found to be
successful only in some low-dimensional cases. As a byproduct of this direct
method, an interesting new integral representation for $\zeta(2)$ is obtained.