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

Abstract

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.

Time-optimal navigation through quantum wind

“…With the advances in the implementation of quantum technologies, the theoretical understanding of controlled quantum dynamics and, in particular, of their limits, is becoming increasingly important. One aspect of such limits that has been investigated extensively in the literature concerns the time-optimal manoeuvring of quantum states [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. If the time-evolution is unconstrained (apart from a bound on the energy resource), then this amounts to finding the time-independent Hamiltonian that generates maximum speed of evolution.…”

Spin nature of genetic code

“…When one of the electrons migrates onto the template (see above), its spin "feels" the hfc effect emerging from three 31 P nuclei. This happens because the template has a right-hand twist sequentially, domain after domain (each of three nucleotides), blocking the hfc effect between the free electron and the phosphorus atoms (as we shall see the effect is directly linked to the value cos)-the gauge field flux associated in our case with B [24,25]. The twist cannot be changed, say to the left twist (the nature will not allow it), or exceed the value of π/2 (a polar angle, ), Figure 3 [26,27].…”

Kinematics of the Rolling Sphere and Quantum Spin

“…The name comes from the fact that, when B 0 = 0 the levels for spin up and down cross at s = 0. The remarkable result obtained by Zener is that the probability of the spin remaining up after the evolution is, in our notation In [5] we show show that the non-adiabatic limit when the levels are crossed very fast (which for rolling corresponds to a sphere much larger than the size of the Cornu spiral) can be obtained in a simple way using the rolling picture. Note that the control problem in this case does not correspond to optimal motion in the sense desribed earlier.…”

“…[12], especially their classical view of the Landau-Zener [14] problem. In a related paper, see [5], we use the rolling analogy to analyze the Lanau-Zener problem. We discuss this idea briefly at the end of this paper and we hope to extend this idea further to the control of spins.…”