Path integrals for the quantum microcanonical ensemble

Abstract

Path integral representations for the quantum microcanonical ensemble are presented. In the quantum microcanonical ensemble, two operators are of primary interest. First, $\stackrel{^}{\ensuremath{\rho}}=\ensuremath{\delta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})$ corresponds to the microcanonical density matrix and can be used to calculate expectation values. Second, $\mathrm{N\ifmmode \hat{}\else \^{}\fi{}}=\ensuremath{\Theta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})$ can give the number of states with energy ${E}_{n}<E.$ We consider position matrix elements of both of these operators $\ensuremath{\Omega}{(x,x}^{\ensuremath{'}},E)=〈{x}^{\ensuremath{'}}|\ensuremath{\delta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})|x〉$ and $\ensuremath{\Theta}{(x,x}^{\ensuremath{'}},E)=〈{x}^{\ensuremath{'}}|\ensuremath{\theta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})|x〉.$ A path integral formalism leads to exact integral representations for $\ensuremath{\Omega}{(x,x}^{\ensuremath{'}},E)$ and $\ensuremath{\Theta}{(x,x}^{\ensuremath{'}},E).$ We present both phase space and configuration space forms. For simple systems, such as the free particle and harmonic oscillator, exact solutions are possible. For more complicated systems, expansion schemes or numerical evaluations are required. A perturbative calculation and numerical integration results are presented for the quantum anharmonic oscillator.