From classical to quantum chaos

You remember last "Physics Friday" post: We showed a phase space map of a billiard in a magnetic fields  as an example for a system showing classical chaos. As in classical Hamiltonian systems, the concept of integrability can also be applied to quantum systems. In the form of quantum numbers conserved quantities in quantum mechanics are related to symmetries. These quantum numbers are the eigenvalues of operators that "generate" the transformation under which the system is invariant (i.e., the operator counterparts of the classical conserved quantities).
The word quantum chaos might let us think of unpredictable behavior in quantum phenomena, but this is not the truth. In fact the solution of the linear Schrödinger equation cannot behave chaotically in the same way as that of quantized classically chaotic systems.

Quantum chaos means a quantum manifestation of chaos in deterministic classical mechanics.(Nakamura in Quantum Chaos and Quantum Dots)

This means the manifestation of chaos is the common characteristic phenomena of quantized classical chaotic systems. One way to see the effect of the classical dynamics is to study local statistics of the energy spectrum, such as the level spacing distribution P(s) which is the distribution function of nearest neighbour spacings as we run over all energy levels. A dramatic insight of quantum chaos is given by the universality conjectures for P(s):

  • If the classical dynamics is integrable, then P (s) coincides with the corresponding quantity for a sequence of uncorrelated levels (the Poisson ensemble) with the same mean spacing.

  • If the classical dynamics is chaotic, then P(s) coincides with the corresponding quantity for the eigenvalues of a suitable ensemble of random matrices.

Level spacing distribution for the energy spectrum of a quantum particle in a circular region vs. the level spacing distribution for Gaussian Unitary Ensemble, Gaussian Orthogonal Ensemble, and Poisson, respectively. Kriecherbauer T et al. PNAS 2001;98:10531-10532

Note that not a single instance of these conjectures is known, in fact there are counterexamples, but the conjectures are expected to hold “generically”, that is unless we have a good reason to think otherwise.