We can define a long-time average of the expectation value of the operator  according to the expression
If we use the explicit expression for the time evolution of this expectation value, we can write
The integration in this expression can be performed explicitly, and the result is
Each of the terms in the second sum will become smaller as the limit is taken to infinity. Assuming that the phase coherence between the different exponential terms in the second sum does not ever become large enough to rival this decay, the second sum will go to zero, and we find that the long-time average of the expectation value is given by
This prediction for the time-average of the observable  is referred to as its predicted value in the diagonal ensemble, The most important aspect of the diagonal ensemble is that it depends explicitly on the initial state of the system, and so would appear to retain all of the information regarding the preparation of the system. In contrast, the predicted value in the microcanonical ensemble is given by the equally-weighted average over all energy eigenstates within some energy window centered around the mean energy of the system
where  is the number of states in the appropriate energy window, and the prime on the sum indices indicates that the summation is restricted to this appropriate microcanonical window. This prediction makes absolutely no reference to the initial state of the system, unlike the diagonal ensemble. Because of this, it is not clear why the microcanonical ensemble should provide such an accurate description of the long-time averages of observables in such a wide variety of physical systems.
However, suppose that the matrix elements  are effectively constant over the relevant energy window, with fluctuations that are sufficiently small. If this is true, this one constant value A can be effectively pulled out of the sum, and the prediction of the diagonal ensemble is simply equal to this value,
where we have assumed that the initial state is normalized appropriately. Likewise, the prediction of the microcanonical ensemble becomes
The two ensembles are therefore in agreement.
This constancy of the values of  over small energy windows is the primary idea underlying the eigenstate thermalization hypothesis. Notice that in particular, it states that the expectation value of  in a single energy eigenstate is equal to the value predicted by a microcanonical ensemble constructed at that energy scale. This constitutes a foundation for quantum statistical mechanics which is radically different from the one built upon the notions of dynamical ergodicity.