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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
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 .
Next: 4.2 Partial occupation of
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Peter Haynes