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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Floyd Williams
    • 1
  1. 1.Department of MathematicsUniversity of MassachusettsAmherstUSA

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