Berry's Phase & Fine Structure Constant Java simulation

Bernd Binder, 9.08.2002, 24.10.2006, 27.04.2010

To visualize the surprisingly simple model of atomic fine structure and spin-orbit coupling in Berry's Phase and Fine Structure (pdf 333kb), a small Java simulation is loaded to this page. Consider a spinning wave packet precessing on a spherical or hyperbolic surface with constant curvature and spin-orbit coupling constant α. Here precession can be assigned to the Berry-Hannay Phase emerging from the spinning or rolling motion on the curved surface (providing for holonomy or anholonomy). The corresponding classical picture is given by epicycloidal or hypocycloidal loops of a spinning top dynamics. It is well known that the free gyroscopic dynamics can be described by a cone rolling in or on cones. The motion of a sphere, rolling and spinning in the interior of a spherical bowl, or on the top of a sphere, has the same character as the motion of the axis of a spinning top about a fixed point. The characteristic linear spin-precession coupling due to the rolling dynamics (precession proportional to tilt and spin) is forced in a class of wide spread mechanical toys (gyrotwister) utilizing the resulting strong moments of precession controlling the spin. Essentially, precession is playing a double role controlling and balancing both, the dynamic and geometric part of the spin/phase evolution. In the simulation below the total phase change jπ is the sum of the Berry/Hannay phase (black segment)

Berry Phase = jπ(1-cosθ)

and the dynamical phase jπMα (purple segment) with precession angle θ = jπα. In the quantum picture the linear spin-precession coupling corresponds to the dynamic phase part given by a linear Hamiltonian spin flux jπMα linearly related to precession by a quantum Dirac monopole charge M. The Berry phase provides for an extra spin flux in addition to the dynamic phase evolution. Consequently, the geometric phase part and missing or extra fraction of wave packet loops in the spin current is

1-Mα = 1-cosθ = 1/N.

As the central result, the major dynamic part is given by the recurrent fixed point condition

Mα = cos(jπα)

Mθ = jπcosθ

Below it is shown how an integral or half-integral N Berry phase fraction can support the fractional revivals of the rolling wave packet. The Java applet shows the epicycloidal (red, positive dynamical phase) and hypocycloidal (blue, negative dynamic phase) rolling dynamics. For both cases the values of 1/α and the ratio dynamical to geometric phase N= cos(jπα)/[1- cos(jπα)] are displayed separated by commas. Moving the slide bar of the Java applet the monopole number M changes (up to 150).

The applet is written in Java. You must have a Java enabled browser to be able to see the applet right here.

The Berry geometric phase (from "parallel transport") is responsible for 1/α-M > 0. So precession and spin-orbit coupling α are induced by a magnetic charge recurrently controlling precession via geometric effects and extra loops. Here one closed orbital loop with one geometric phase loop is devided or quantized into M equal angular monopole units or segments (free loop condition) or an integral number N of equal epicycloidal or hypocycloidal closed rolling loops. A quantum precession and path/loop condition can be assigned to the integral winding numbers N and/or M. In the classical picture this could be visualized as a rolling cone resonance conditon with rolling plane periodically tilted at precessing angle θ. The recurrent relation for a given M is in chaos theory known as the cosine map that can be solved iteratively. Therefore, the irrational number α resulting from an iterative process of a geometric object as a pure geometric feedback-loop condition is due to a linear spin-precession coupling. Now the low frequency precession can control and drive the high frequency spin loops near the Magic Angle Precession (MAP) fixed point α resulting in a rather strong coupling between the spin and the curved embedding space responsible for precession.
The source of this applet is here. The radius ratio of the two neighboring red and blue rings in the "free" case is cos(jπα). Here j=1, where j is spin times charge, so the situation could describe a binary rolling dynamics of a Cooper pair with half spin times double charge. The physics behind α can be modelled by holonomy relations in classical mechanics (rolling cones or balls, Hannay, Pancharatnam, can be experienced with a Dynabee or Gyrotwister sports/fitness device) and fits to a general spin-orbit coupling relation. The coupling is polar (sign of j) and carries a Dirac monopole with half spin magnetic charge M/2 on SU(2)/U(1) = S^2. Assigned to a physical entity α is the number of precession loops (orbital precession wavenumber) divided by the number of wave packet loops (Compton wavenumber). The value 1/α=137.0359999 ... (M=137) obtained from a free running iteration can be considered as a candidate for the inverse Sommerfeld fine structure constant in agreement with recent measurements 1/α=137.035 999 45 (62), see here. It should be possible (but could be very hard) to observe the substructure given by a variable integral N supporting fractional revival.

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