The rolling sphere, the quantum spin and a simple view of the Landau-Zener problem

We consider the problem of a sphere rolling on a curved surface and solve it by mapping it to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. The mapping can be of pedagogical use in discussing both rolling and spin precession. As an interesting example we show that the Landau-Zener problem corresponds to the rolling of a sphere on a Cornu spiral, and derive the probability of a non-adiabatic transition using the rolling language. We also discuss the adiabatic limit and the vanishing of geometric phases for rolling on curved surfaces.