In section 2.1 we laid down the fundamental principles of quantum mechanics in terms of wave-functions and operators. In practice, however, we often do not know the precise quantum-mechanical state of the system, but have some statistical knowledge about the probabilities for the system being in one of a set of states (note that these probabilities are completely distinct from the probabilities which arise when a measurement is made). For a fuller discussion of what follows, see [130].

Suppose that there is a set of orthonormal states
for our system, and that the probabilities that the
system is in each of these states are . The expectation
value of an observable is

(4.1) |

We define the * density-operator* as

(4.2) |

(4.3) |

(4.4) |

in which the density-matrix , the matrix representation of the density-operator in this basis, is defined by

(4.5) |

The fact that the probabilities must sum to unity is expressed by the
fact that the trace of the density-matrix is also unity i.e.
.
A state of the system which corresponds to a single state-vector (i.e. when
and
) is known as a * pure state*
and for such a state the density-matrix obeys a condition known as
* idempotency* i.e. which is only obeyed by matrices
whose eigenvalues are all zero or unity. The more general state introduced
above is known as a * mixed state* and does not obey the idempotency
condition. Other properties of the density-matrix are that it is Hermitian,
and that in all representations the diagonal elements are always real and
lie in the interval .