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