Density Operator Formalism

The occupation probabilities of energy levels of a macroscopic system in thermodynamic equilibrium are independent of time. The formalism of equilibrium statistical mechanics we presented could therefore be based on the occupation probabilities of energy levels. Those probabilities, however, change in time as the system evolves starting from an arbitrary initial state. That change takes place due to transitions between the energy levels. Hence time evolution of a system cannot be described only in terms of the probabilities of occupation of energy levels. In other words the probabilities of occupation of energy levels do not characterize the state of the system completely. A complete description of the state must include transition probabilities as well. We know that the state of an isolated system is characterized completely by a vector in the Hilbert space. The systems of interest in statistical mechanics are, however, not isolated; they interact with reservoirs. In this chapter we develop the formalism to characterize the state of an interacting system, called the density matrix formalism.