Statistical Mechanics - Physics at Oregon State University

6.1. DENSITY OPERATORS. 121
The partition function is
Z =
n
pn = 1 En
k e− B T (6.8)
Z
En
−
e
k B T =
n
En
− k 〈n| e B T |n〉 (6.9)
Now we use the fact that the states in the sum are eigenstates of the Hamiltonian,
and we obtain
Z =
n
H − k 〈n| e B T −
|n〉 = Tr e H
kB T (6.10)
At this point we will switch to the more common notation for the inverse
. The density operator is given by
temperature, β = 1
kBT
ρ = 1
Z
e −βEn |n〉 〈n| (6.11)
n
and using the fact again that we expand in eigenstates of the Hamiltonian:
ρ = 1
Z e−βH
(6.12)
These definitions are basis-dependent. The only question is how to define the
exponent of an operator, but that turns out to be easy since the power series
for exponents always converge uniformly on any finite interval. Hence the ex-
ponent of an operator O is defined by e O = ∞
n=0 1
n! On . The mathematical
filed of functional analysis, very useful for physicists, gives many more details.
Products of operators are well defined. Therefore, the density operator can be
specified without knowing the eigenstates of the Hamiltonian. This is definitely
an advantage. Note, however, that in many cases the most practical manner
to do calculations is by using eigenvalues. But there are certainly other cases
where it is better not to do so. Finally, the expectation value of an operator A
in this formalism is
< A >= Tr (ρA) = Tr (e−βH A)
Tr (e −βH )
(6.13)
This is an important form for expectation values. It can be used in diagrammatic
techniques in the same manner as Feynman diagrams. The only substitution
we need to make is β → ıt. We replace inverse temperature by imaginary time.
Now what does that mean???
Formulas (6.10) and (6.12) contain functions of operators. Elementary functions
of an operator are positive integral powers of an operator. For example,
O 2 |x〉 = O |y〉 with |y〉 = O |x〉. Sums and differences of operators are also
easily defined. More complicated functions of operators can be defined through
power series, just like functions of a complex variable in complex analysis. These
power series often have a certain radius of convergence, and expressed in terms