Quantum Chaos, Level spacing distr. in integrable system

[Scipio]

Hello all,

For an undergraduate essay, I am studying the development of quantum chaos in a 1D spin 1/2 chain (my main source paper can be found here:http://scitation.aip.org/content/aapt/journal/ajp/80/3/10.1119/1.3671068).

One of the main tools used to distinguish chaotic from non chaotic systems is the level spacing distribution. According to this paper and all other sources I have read, including wikipedia, "the energy levels of integrable systems or not correlated, and are not prohibited from crossing, so the distribution is Poissonian" (ie. negative exponential), "In chaotic systems the eigenvalues become correlated and crossings are avoided. There is level repulsion, and the level spacing distribution is given by the Wigner Dyson distribution", which is similar to the distribution obtained with the spectra of random matrices.

However, many simple systems (infinite square well, harmonic oscillator) have clearly defined functions for the eigenenergies (~n^2, ~n, respectively). If you calculate the spacing between these states and create a histogram out of these, you don't get anything like a Poissonian distribution (for the harmonic oscillator, the distance between all levels are, of course, equal, so the histogram is not very interesting; for the infinite square well, I obtain a completely homogeneous distribution for the spacing of the first 100 eigenvalues, using E[n] = n^2 and a bin width of 10). The first simple system where I obtain the Poissonian distribution is a particle in a box. Are these other simple systems just exceptions to the rule? Is there some intuitive argument to see why?

Thank you!

Ps. cross posted with http://physics.stackexchange.com/questions/151042/quantum-chaos-level-spacing-distribution-in-integrable-quantum-systems

 

[atyy]

Yes, the other simple systems like the harmonic oscillator are exceptions to the conjectured rule. So one has to add some appropriate definition of "generic" to the conjecture. You could try http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/marklof.pdf to see if anyone has proven some more precise version of the Poisson half of the conjecture.