At equilibrium, the density operator does not evolve in time; thus,
. Thus, from the equation of motion,
if this holds, then 
, and 
is a constant of the
motion. This means that it can be simultaneously diagonalized with
the Hamiltonian and can be expressed as a pure function of the
Hamiltonian
Therefore, the eigenstates of 
, the vectors, we called 
are the eigenvectors 
of the Hamiltonian, and we can write
H and as
The choice of the function f determines the ensemble.