Statistical Mechanics - Physics at Oregon State University

122 CHAPTER 6. DENSITY MATRIX FORMALISM.
of operators are only valid if the norm of the operator is less than this radius of
convergence. But often these functions can be continued for arbitrary operators
similar to the analytic continuation procedure in complex analysis. For example,
e O is defined via a power series for all operators O. If we can solve the operator
equation O = e X for the operator X, then we have defined X = log(O). The
logarithm of an operator can be expressed in the form of a power series for
log(1 + O) only if the norm of O is less than one. But log(1 + O) is defined for
all operators O for which O = e X can be solved; this is a much larger class of
operators.
If an operator O is Hermitian, O = O † , the definition of functions of operators
can be related to standard functions of real variables. A Hermitian
operator has a complete set of eigenfunctions |n〉 with corresponding eigenvalues
λn. If f(x) is a well-defined function a real variable x in a certain range,
the definition of f can be extended to operators by O = λn |n〉 〈n| ⇒ f(O) =
f(λn) |n〉 〈n|. For example, the logarithm of a Hermitian operator can be
defined this way only if none of the eigenvalues of the operator is zero! In other
words, O = e X with O Hermitian can only be solved if none of the eigenvalues
of O is zero.
This treatment allows us to check what happens for an operator with ρ 2 ρ.
Suppose we have
This gives
ρ |n〉 = rn |n〉 (6.14)
〈n| ρ 2 |n〉 〈n| ρ |n〉 ⇒ r 2 n rn
(6.15)
which means that rn is not negative and at most equal to one, 0 rn 1.
Hence for an arbitrary vector:
〈x| ρ 2 |x〉 =
〈x| ρ |n〉 〈n| ρ |x〉 =
〈x| rn |n〉 〈n| rn |x〉 =
r 2 n| 〈n |x〉 | 2
Similarly:
n
〈x| ρ |x〉 =
〈x| ρ |n〉 〈n |x〉 =
〈x| rn |n〉 〈n |x〉 =
rn| 〈n |x〉 | 2
n
and since r2 n rn we have
r 2 n| 〈n |x〉 | 2
rn| 〈n |x〉 | 2
n
n
n
n
n
n
(6.16)
(6.17)
(6.18)
or 〈x| ρ 2 |x〉 〈x| ρ |x〉 for all possible vectors. Hence the definition for ρ 2 ρ
is a valid one. Once we define this condition on all eigenstates, it is valid
everywhere. This is another example showing that it is sufficient to define
properties on eigenstates only.