^{1,a)}and Anthony M. Bloch

^{2,b)}

### Abstract

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.

The authors thank Sir Michael V. Berry for useful comments on the manuscript and for pointing the authors to Ref. 5. The authors thank Roger Brockett and Paul R. Berman for interesting remarks. The authors would also like to thank Gil Bor and Richard Montgomery for helpful remarks on inner and outer rollings and Gil Bor for pointing out a crucial sign error in an earlier version of this paper. The authors thank the referees for their careful review of the manuscript. A.G.R. thanks the Research Corporation, and A.M.B. thanks the National Science Foundation for support.

I. INTRODUCTION

II. THE PHYSICS OF SPIN

III. ROLLING ON A PLANE AND QUANTUM PRECESSION

IV. WARMUP: CONSTANT MAGNETIC FIELD

V. THE LOLLIPOP AND THE PLANAR FIELD

VI. ROLLING ON A CORNU SPIRAL AND THE LANDAU–ZENER PROBLEM

A. Landau–Zener expression in rolling language

VII. ROLLING ON A CURVED SURFACE

VIII. SPHERE ROLLING ON A SPHERICAL SURFACE

IX. THE ADIABATIC APPROXIMATION AND ROLLING ON A CURVED SURFACE

### Key Topics

- Magnetic fields
- 25.0
- Adiabatic theorem
- 9.0
- Geometric phases
- 8.0
- Non adiabatic reactions
- 8.0
- Eigenvalues
- 7.0

## Figures

A sphere rolling on a straight horizontal line corresponds to a spin 1/2 in a constant magnetic field in the direction.

A sphere rolling on a straight horizontal line corresponds to a spin 1/2 in a constant magnetic field in the direction.

The lollipop, or a sphere rolling counterclockwise on a circle of radius , corresponds to a spin 1/2 precessing in a magnetic field that rotates in the plane.

The lollipop, or a sphere rolling counterclockwise on a circle of radius , corresponds to a spin 1/2 precessing in a magnetic field that rotates in the plane.

Equivalence of (a) rolling on a Cornu spiral and (b) the Landau–Zener problem of the spin flip probability on a time dependent field.

Equivalence of (a) rolling on a Cornu spiral and (b) the Landau–Zener problem of the spin flip probability on a time dependent field.

Sphere rolling along a curve of zero torsion (meaning that the velocity of the center of the sphere is parallel to the tangent of the curve at the contact point).

Sphere rolling along a curve of zero torsion (meaning that the velocity of the center of the sphere is parallel to the tangent of the curve at the contact point).

When a sphere of radius rolls on the parallel of a second sphere of radius , the angular velocity describes a cone. In the mapping from rolling spheres to spins, the magnetic field also describes a cone. This case of a sphere rolling on a parallel is isomorphic to a spin 1/2 precessing in a time dependent magnetic field and describing a cone at constant rate.

When a sphere of radius rolls on the parallel of a second sphere of radius , the angular velocity describes a cone. In the mapping from rolling spheres to spins, the magnetic field also describes a cone. This case of a sphere rolling on a parallel is isomorphic to a spin 1/2 precessing in a time dependent magnetic field and describing a cone at constant rate.

The instantaneous position of two spheres of radius rolling on a sphere of radius . One sphere rolls outside and the other inside the sphere of radius . The plane of the paper corresponds to the instantaneous plane containing the centers of the spheres and the angular frequencies and for inner and outer rollings.

The instantaneous position of two spheres of radius rolling on a sphere of radius . One sphere rolls outside and the other inside the sphere of radius . The plane of the paper corresponds to the instantaneous plane containing the centers of the spheres and the angular frequencies and for inner and outer rollings.

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