NONLINEAR DYNAMICS: QUANTUM CHAOS ----------------------------------------------------------- Georgia Tech PHYS 7224 Spring semester 2002 Predrag Cvitanovic' Lecture 24 Tue, Apr 9 2002 12:05-13:25 in Howey S106 Helium, classical dynamics ---------------- In continuing episode of indomitable optimism I continue discussing classical dynamics of helium - symmetries, symbolic dynamics. Reading: -------- Classical and Quantum Chaos --- www.nbi.dk/ChaosBook/postscript.html chapter 24 - Helium atom 24.1.4 Local coordinates, Jacobian matrix chapter 4 - Local stability (optional) chapter 12 - Fixed points, and how to get them (optional) chapter 17 - Discrete symmetries 17.5 C_2 factorization Exercises for chapter 24 Problem set ----------- Exer 17.A C_2 factorization derive relations between fundamental domain and full space cycles listed in Table 17.1. Due Thu 11 2002. Rest due Tue 16 2002: Exer 24.5 Collinear helium cycles. Guess some short cycles by requiring that topologically they correspond to sequences of bounces either returning to the same r_i axis or reflecting off the diagonal. Compute them by some method or other. (if you are quantizing 3-disks rather than helium, find all cycles up in fundamental domain to length 5.) Exer 24.6 Collinear helium cycle stabilities Compute the stability eigenvalues for you collinear helium cycles. Compare with Table 24.1 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. -------------------------------------------------------------------