Berry's Phase & Fine Structure Constant Java simulation

Bernd Binder, 9.08.2002, 24.10.2006, 8.02.2010

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 rolling on a curved spherical surface due to a radial coupling. The emerging Berry geometric phase fraction jπ[1-cos(jπα)] (driven and controlled by a precession angle θ = jπα) and the corresponding spin flux as the missing fraction of wave packet rolling loops jπ(1-Mα) = jπ/N balance the spin-orbit coupling constant α = cos(jπα)/M as a fixed point. The recurrent relation is in chaos theory known as the cosine map that can be solved iteratively. The resulting irrational number α can be assigned to a physical entity based on an iterative process of a geometric object as a pure geometric feedback-loop due to a spin-precession coupling, where α is the number of precession loops (orbital precession wavenumber) divided by the number of wave packet loops (Compton wavenumber) and M the integral Dirac magnetic monopole quantum number. Due to the spin-precession coupling the low frequency precession can control and drive the high frequency spin near the Magic Angle Precession (MAP) fixed point resulting in a rather strong coupling between the spin and the curved embedding space recurrently establishing spin-orbit coupling. Embedded in hyperbolic space the geometric phase anholonomy or rational angular defect (N-j)/N leading to precession (in relativity a Thomas precession) acts like "contracting" not only the azimuthal (0...2π) but also the zenithal angular range (0...π).
As the key result, the Berry geometric phase (from "parallel transport") is responsible for 1/α-M > 0, where the total phase change jπ is the sum of the Berry phase jπ[1-cos(jπα)] and the dynamical phase jπMα. Moving the slide bar of the Java applet changes M 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 source of this applet is here. The Java applet shows the epicycloidal (red, positive dynamical phase) and hypocycloidal (blue, negative dynamical phase) rolling dynamics. For both cases the values of 1/α and the ratio dynamical to geometric phase N= jcos(jπα)/[1- cos(jπα)] are displayed separated by commas (with integer or half-integer N the Berry phase can support the fractional revivals of the rolling wave packet). 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 M) and carries a Dirac monopole with half spin magnetic charge M/2 on SU(2)/U(1) = S^2. 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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