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