The Density Matrix and Partition Function in Quantum Statistical Mechanics

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


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:
$$\begin{array}{*{20}{c}} {\rho {{{(\beta ;{{x}_{a}},{{x}_{b}})}}^{{\underline{\underline {def}} }}}\int_{{{{x}_{a}} = x(0)}}^{{{{x}_{b}} = x(\beta \hbar )}} {\exp ( - {{S}_{E}}(x(t))/\hbar )D[x(t)]} } \\ { = {{I}_{E}}(0,\beta \hbar ,{{x}_{a}},{{x}_{b}})} \\ \end{array}$$
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:
$$\begin{array}{*{20}{c}} {Z = \int_{{ - \infty }}^{\infty } {\rho (\beta ;x,x)dx} } \hfill \\ { = \int_{{ - \infty }}^{\infty } {\int_{{x(0) = x}}^{{x(\beta \hbar ) = x}} {\exp ( - {{S}_{E}}(x(t))/\hbar )D[x(t)]dx.} } } \hfill \\ \end{array}$$


Free Energy Partition Function Density Matrix Internal Energy Zeta Function 
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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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