Quantum Precession (Magic Angle Precession in curved spacetime)

Quantum Precession or Quantum Gyroscope with Magic Angle Precession (MAP)
(date: Okt.-Nov.2006, small text and simulation update 2.6.2008, 3d Java simulation)

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Basics:
Consider a situation in quantum mechanics where gravitational, gravitomagnetic, gravitoelectric, or electromagnetic interaction affects the spin quantum numbers. A device that senses this effect (curvature or vector potentials) via rotation, precession, and nutation is a Quantum Gyroscope. Nutations are spherical cycloids (epicyclical or hypocycloidal loops) with corresponding spherical harmonics in the quantum case. Let N be the integral number of nutations for one rotation (the azimuthal or longitudinal spin quantum number), where N and N-j are the numbers in flat space and curved space, respectively. The missing nutation period can be assigned to a relativistic effect (i.e. the "hyperbolic defect" of angles) that can be interpreted as a quantum geometric phase pj/N, a precession quantum induced by curvature intimately related to Berry's geometric phase. Gyroscopic precession can be modeled by a rolling cone. In classical mechanics the solid-body rotation angle is connected to the actual solid angle that the body-fixed axis describes as the body performs a conical motion (Ishlinskii, Malykin). The cone apex angle is p(1-ja)/2, while the zenithal (polar or colatitudinal) precession angle jpa/2 is given by

cos(jpa) = (N-j)/N.

A fast dynamics can be related to a special relativistic effect known as Thomas precession where the "hyperbolic defect" is responsible for the Thomas angle. This extra or rather missing space rotation is the angular defect or "contraction" in the hyperbolic triangle.

New:
We assume that the angular defect is isotropically "contracting" not only the azimuthal (0...2p) but also the zenithal range (0...p) down by a rational factor (N-j)/N. Without curvature defect the quantum precession angle theta = pja would be pj/M with integral zenithal quantum number 0 < M < N . But since the precession angle jpa is also contracted by (N-j)/N it couples back to the contracted azimuthal range (number of nutations) resulting in the balancing relation and chaotic attractor (showing bifurcations for j/M>0.86...)

cos(jpa) = Ma , a ~ (N-j)/N /M .

This relation with fixed M and running N determines a and can be solved by iteration (in the simulation above: M = 13, j=1, a = 1/13.3.., N = 36). As a general relativistic model of spin-orbit coupling the magic precession angle for electromagnetic interaction (Sommerfeld, Dirac) is with M = 137 very accurately given by a = 1/137.03600.., N = 3806. Similar to Magic Angle Spinning (MAS) in NMR it provides for spin resonance with very narrow spectral lines. The rolling cone model is not only a nice visualization it is an exact model for precession and nutation. Additionally, elliptic Kepler orbits and the corresponding precession of the perihelion can be found by slicing the proper cone dynamics with a stationary plane. Charge Z and spin J with JZ=j can enter this model as additional factors multiplying the precession angle leading to chaos, where the first bifurcation instability occurs for Z>115.05.... for J = 1/2. Find more about this model in

Geometric Phase Locked in the Nucleus (pdf 279kb) and Berry's Phase and Fine Structure (pdf 333kb).

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