 The second question starts with a quantum system that has some relationship with a classical chaotic system via an appropriate semi-classical limit. The classical-to-quantum direction often follows the pioneering work of Martin Gutzweiller (1971) in quantizing the classical chaotic system. The quantum-to-classical direction is much more difficult and fraught with conceptual problems. Standard approaches, here, are to start with a quantum analogue to a classical chaotic system and then derive a semi-classical system that represents the quantum system in some kind of classical limit (Berry 1987 and 2001; Bokulich 2008). This work results in statistics of suitably normalized energy levels for the semi-classical systems with universal features. For classical systems that behave non-chaotically, the energy levels of the semi-classical system approximate a Poisson distribution, where small spacings dominate. In contrast when the classical system behaves chaotically, the energy levels of the semi-classical system take on a distribution originally derived by Eugene Wigner (1951) to describe nuclear energy spectra (for discussion see Guhr et al. 1998). These latter distributions depend only on some symmetry properties (e.g., the presence or absence of time-reversal symmetry in the  system).[11]  Moreover, the presence of periodic orbits in the analog classical systems largely determine the properties of semi-classical systems (Berry 1977). 