NONLINEAR DYNAMICS: QUANTUM CHAOS
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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
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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:
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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
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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.
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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.
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