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

We solve the problem of a sphere rolling on a curved surface by mapping it to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. The mapping can be instructive for discussing both rolling and spin precession. As an 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 nonadiabatic transition using this correspondence. We also discuss the adiabatic limit and the vanishing of geometric phases for rolling on curved surfaces.