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# 4.1 The density-matrix

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)

which is a quantum and statistical average.

We define the density-operator as

 (4.2)

and introduce a complete set of basis states , writing the as linear combinations:
 (4.3)

Expressed in terms of this basis, the expectation value becomes
 (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 .

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Peter Haynes