Quantum ergodicity

Since a classically chaotic system is also ergodic, almost all of its trajectories eventually explore uniformly the entire accessible phase space.

The quantum ergodicity theorems of Shnirelman, Zelditch, and Yves Colin de Verdière proves that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average.

In particular, the theorem allows the existence of a subset of macroscopically nonergodic states which on the other hand must approach zero measure, i.e., the contribution of this set goes towards zero percent of all eigenstates when

[1][5][6][2] The instability is a decisive point that separates quantum scars from a more trivial finding that the probability density is enhanced near stable periodic orbits due to the Bohr's correspondence principle.

Conventional and perturbation-induced scars are both a striking visual example of classical-quantum correspondence and of a quantum suppression of chaos (see the figure).

In particular, scars are a significant correction to the assumption that the corresponding eigenstates of a classically chaotic Hamiltonian are only featureless and random.

In some sense, scars can be considered as an eigenstate counterpart to the quantum ergodicity theorem of how short periodic orbits provide corrections to the universal random matrix theory eigenvalue statistics.

The eigenmode of a classically integrable system (e.g. the circular cavity on the left) can be very confined even for high mode number. On the contrary the eigenmodes of a classically chaotic system (e.g. the stadium-shaped cavity on the right) tend to become gradually more uniform with increasing mode number.
A scar in a stadium billiard (upper panel) and in a disordered quantum dot (lower panel) are example of nonergodic eigenstates allowed by the quantum ergodicity theorem. In both cases, the probability density of the eigenstates is concentrated along a periodic orbit of the classical counterpart (solid blue line). Although similar in appearance, the birth mechanism of the scars in the billiard and the quantum dot perturbed by potential bumps (red dots) are different: the former is explained the conventional scar theory, [ 1 ] [ 2 ] whereas the latter is known as perturbation-induced scarring [ 3 ] [ 4 ] (for further information, see Quantum scar ).