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.
-------------------------------------------------------------------