In statistical mechanics, the microcanonical ensemble is a particular statistical ensemble which is used to make predictions about the outcomes of experiments performed on isolated systems that are believed to be in equilibrium with an exactly known energy. The microcanonical ensemble is based upon the assumption that, when such an equilibrated system is probed, the probability for it to be found in any of the microscopic states with the same total energy have equal probability. With this assumption, the ensemble average of an observable quantity is found by averaging the value of that observable  over all microstates  with the correct total energy: 
Importantly, this quantity is independent of everything about the initial state except for its energy.
The assumptions of ergodicity are well-motivated in classical mechanics as a result of dynamical chaos, since a chaotic system will in general spend equal time in equal areas of its phase space. If we prepare an isolated, chaotic, classical system in some region of its phase space, then as the system is allowed to evolve in time, it will sample its entire phase space, subject only to a small number of conservation laws (such as conservation of total energy). If one can justify the claim that a given physical system is ergodic, then this mechanism will provide an explanation for why statistical mechanics is successful in making accurate predictions. For example, the hard sphere gas has been rigorously proven to be ergodic.
This argument cannot be straightforwardly extended to quantum systems, even ones that are analogous to chaotic classical systems, because time evolution of a quantum system does not uniformly sample all vectors in Hilbert space with a given energy. Given the state at time zero in a basis of energy eigenstates
the expectation value of any observable  is
Even if the  are incommensurate, so that this expectation value is given for long times by
the expectation value permanently retains knowledge of the initial state in the form of the coefficients .
In principle it is thus an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system. However, a variety of experiments in cold atomic gases have indeed observed thermal relaxation in systems which are, to a very good approximation, completely isolated from their environment, and for a wide class of initial states. The task of explaining this experimentally observed applicability of equilibrium statistical mechanics to isolated quantum systems is the primary goal of the eigenstate thermalization hypothesis.