text: Suppose that we are studying an isolated, quantum mechanical many-body system. In this context, "isolated" refers to the fact that the system has no (or at least negligible) interactions with the environment external to it. If the Hamiltonian of the system is denoted , then a complete set of basis states for the system is given in terms of the eigenstates of the Hamiltonian,
  where  is the eigenstate of the Hamiltonian with eigenvalue . We will refer to these states simply as "energy eigenstates." For simplicity, we will assume that the system has no degeneracy in its energy eigenvalues, and that it is finite in extent, so that the energy eigenvalues form a discrete, non-degenerate spectrum (this is not an unreasonable assumption, since any "real" laboratory system will tend to have sufficient disorder and strong enough interactions as to eliminate almost all degeneracy from the system, and of course will be finite in size). This allows us to label the energy eigenstates in order of increasing energy eigenvalue. Additionally, consider some other quantum-mechanical observable , which we wish to make thermal predictions about. The matrix elements of this operator, as expressed in a basis of energy eigenstates, will be denoted by
  We now imagine that we prepare our system in an initial state for which the expectation value of  is far from its value predicted in a microcanonical ensemble appropriate to the energy scale in question (we assume that our initial state is some superposition of energy eigenstates which are all sufficiently "close" in energy). The eigenstate thermalization hypothesis says that for an arbitrary initial state, the expectation value of  will ultimately evolve in time to its value predicted by a microcanonical ensemble, and thereafter will exhibit only small fluctuations around that value, provided that the following two conditions are met:
  # The diagonal matrix elements  vary smoothly as a function of energy, with the difference between neighboring values, , becoming exponentially small in the system size.
  # The off-diagonal matrix elements , with , are much smaller than the diagonal matrix elements, and in particular are themselves exponentially small in the system size.
  These conditions can be written as
  where  and  are smooth functions of energy,  is the many-body Hilbert space dimension, and  is a random variable with zero mean and unit variance. Conversely if a quantum many-body system satisfies the ETH, the matrix representation of any local operator in the energy eigen basis is expected to follow the above ansatz.