Currently, there is no known analytical derivation of the eigenstate thermalization hypothesis for general interacting systems. However, it has been verified to be true for a wide variety of interacting systems using numerical exact diagonalization techniques, to within the uncertainty of these methods. It has also been proven to be true in certain special cases in the semi-classical limit, where the validity of the ETH rests on the validity of Shnirelman's theorem, which states that in a system which is classically chaotic, the expectation value of an operator A ^ ^ in an energy eigenstate is equal to its classical, microcanonical average at the appropriate energy. Whether or not it can be shown to be true more generally in interacting quantum systems remains an open question. It is also known to explicitly fail in certain integrable systems, in which the presence of a large number of constants of motion prevent thermalization.