NONLINEAR DYNAMICS: QUANTUM CHAOS ----------------------------------------------------------- Georgia Tech PHYS 7224 Spring semester 2002 Predrag Cvitanovic' Lecture 27 Thu, Apr 18 2002 12:05-13:25 in Howey S106 Helium, cycle expansions ------------------------ Looking for closure - assuming you have the cycles, you are now ready to determine the collinear spectrum by implementing the the cycle expansions for the symmetric, antisymmetric subspaces. Reading: -------- Classical and Quantum Chaos --- www.nbi.dk/ChaosBook/postscript.html chapter 24 - Helium atom 24.2 Semiclassical quantization of collinear helium Exercises for chapter 24 ------------------------ due Tue 23 2002: Exer 24.7 Collinear helium spectrum. Compute the lowest eigenenergies of singlet and triplet states of helium by substituting cycle data (either your own, or from table 24.1) into the cycle expansions for the symmetric and the antisymmetric zeta functions. Probably the quickest way is to plot the magnitude of the zeta function as function of real energy and look for the minima. As the eigenenergies in general have a small imaginary part, a contour plot such as fig 13.1 can yield informed guesses. Better way would be to find the zeros by Newton method. How close are you to the cycle expansion and quantum results listed in table 24.2? (if you are quantizing 3-disks rather than helium, use all cycles up to the fundamental domain length 5) Perhaps some of the helium stuff on http://www.nbi.dk/ChaosBook/extras might help you. If you have developped some half-way decent programs, comment them richly and contribute to ChaosBook extras homepage. -------------------------------------------------------------- The solution of problems starting with these three and ending with the helium spectrum form the final exam in the course. If you prefer, you can compute the 3-disk resonances instead, the determination of perioidc orbits is easier in billards than it is in the flows. -------------------------------------------------------------------