Progress in mathematical physics

The Density Matrix and Partition Function in Quantum Statistical Mechanics

Floyd L. Williams

Published Jan 1, 2003• 0 citations

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)

DOI: 10.1007/978-1-4612-0009-3_15

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