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


Image of Fig. 1.
Fig. 1.

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

Image of Fig. 2.
Fig. 2.

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.

Image of Fig. 3.
Fig. 3.

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

Image of Fig. 4.
Fig. 4.

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).

Image of Fig. 5.
Fig. 5.

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.

Image of Fig. 6.
Fig. 6.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: The rolling sphere, the quantum spin, and a simple view of the Landau–Zener problem