Random a priori phase

Construction of the density operator can also not be achieved without assumption of an additional axiom All quantum states of a system compatible with the knowledge revealed by macroscopic measurement have equal a priori probabilities and random a priori phases. This axiom implies that for a system as defined above all diagonal elements of the density matrix q belonging to the ith cell must be equal. Hence... 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

In this energy regime, the condition of the system is correctly represented not by any one state , M but rather by a quantum statistical ensemble in which all the accessible states appear with equal a priori weights and random a priori phases. The appropriate density operator is then... 

A more complete version of the second postulate assumes random a priori phases of the time-dependent factors in the wave functions and involves quantum-mechanical density matrices. We present only the simplest version of statistical mechanics. 

---