Statistical Mechanics - Physics at Oregon State University

120 CHAPTER 6. DENSITY MATRIX FORMALISM.
< A >=
pn 〈n| A |n〉 (6.2)
n
The values of pn are all we know about the state of our system, and they
give a complete description of this state. In this case the description of the
system is an ensemble, which is a more general way to describe a system than
using pure states only. Of course we need pn 0 and
n pn = 1. A pure state
corresponds to pm = 1 for a particular value of m. We now define the operator
ρ by
ρ =
pn |n〉 〈n| (6.3)
n
This operator is called a density operator, and the matrix of its elements in a
given representation is called a density matrix. The terms operator and matrix
are frequently used for both, however. The operator ρ obeys the following
relations:
Tr ρ =
〈n| ρ |n〉 =
pn = 1 (6.4)
n
ρ = ρ †
n
(6.5)
ρ 2 ρ (6.6)
The last equation has to be interpreted in the following manner: we have
〈x| ρ 2 |x〉 〈x| ρ |x〉 for all states |x〉. A consequence of this equation is that
ρ is non-negative, 〈x| ρ |x〉 0 for all states |x〉. A Hermitian operator is defined
by 〈x| ρ † |y〉 = (〈y| ρ |x〉) ∗ for all states |x〉 and |y〉. The equations (6.4)
through (6.6) can also be used as a definition of a density operator. Any operator
obeying these requirements is a density operator of the form (6.3) and the
probabilities pn are the eigenvalues of the density operator.
If we know the density matrix for a system, we know everything about this
system. This is due to the fact that we can only measure expectation values of
observable quantities, and these expectation values are obtained from a density
matrix by
< A >=
〈n| pnA |n〉 = Tr (ρA) (6.7)
n
where we have used the fact that ρ |n〉 = pn |n〉 for the basis set of eigenstates
of the density operator. A big advantage of this formulation is that it is basis
independent. For example, the trace of a matrix is independent of the choice
of basis states. The density matrix can be specified in any basis, or in operator
form.
In statistical mechanics the probabilities pn are equal to the Boltzmann
factor, and we have