According to the first postulate of quantum mechanics, the state of an isolated system is described by a vector in the Hilbert space. However, more often than not, we encounter systems interacting with other systems. In this chapter we address the question of describing the state of a system interacting with other systems without any reference to the details of the other system or that of the interaction between them. This leads to the concept of the density operator. The concept of density operator will be seen to arise also while characterizing the state of a mixture of particles in different states.

State of a Subsystem

Consider a system composed of the subsystems A and B lying in the Hilbert space of dimensions nA and nB, respectively. We use the formalism developed in Section 5.6 for describing a composite system. Let the state of the composite system be. Recall (5.71) which expresses the composite state of A and B in terms of the superposition of the tensor product of their basis states:

where and are the sets of the orthonormal basis states of A and B. Let it be that we are interested in the properties of only one of the subsystems, say, the subsystem A. Those properties are determined by the expectation values of the operators which act on the states of the subsystem A alone.