# The Density Matrix and Partition Function in Quantum Statistical Mechanics

• Floyd Williams
Chapter
Part of the Progress in Mathematical Physics book series (PMP, volume 27)

## Abstract

In quantum statistical mechanics one has the fundamental object ρ(β; x a ,x b ), a matrix entry which defines the density matrix ρl, given in the Feynman formulation by the Euclidean path integral of the preceding chapter:
$$\begin{array}{*{20}{c}} {\rho {{{(\beta ;{{x}_{a}},{{x}_{b}})}}^{{\underline{\underline {def}} }}}\int_{{{{x}_{a}} = x(0)}}^{{{{x}_{b}} = x(\beta \hbar )}} {\exp ( - {{S}_{E}}(x(t))/\hbar )D[x(t)]} } \\ { = {{I}_{E}}(0,\beta \hbar ,{{x}_{a}},{{x}_{b}})} \\ \end{array}$$
(14.1.1)
where $$\beta = \tfrac{1}{{kT}}$$ is the inverse temperature $$\tfrac{1}{T}$$ up to a constant $$\tfrac{1}{k}$$, k being Boltzmann’s constant. From ρ one obtains the all-important partition function Ζ as a trace:
$$\begin{array}{*{20}{c}} {Z = \int_{{ - \infty }}^{\infty } {\rho (\beta ;x,x)dx} } \hfill \\ { = \int_{{ - \infty }}^{\infty } {\int_{{x(0) = x}}^{{x(\beta \hbar ) = x}} {\exp ( - {{S}_{E}}(x(t))/\hbar )D[x(t)]dx.} } } \hfill \\ \end{array}$$
(14.1.2)

## Keywords

Free Energy Partition Function Density Matrix Internal Energy Zeta Function
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