where | E α α ⟩ ⟩ is the eigenstate of the Hamiltonian with eigenvalue E α α . 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 A ^ ^ , 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