A recently developed method systematically improved the convergence of generic path integrals for transition amplitudes, partition functions, expectation values and energy spectra. This was achieved by analytically constructing a hierarchy of discretized effective actions indexed by a level number p and converging to the continuum limit as 1/N p . Here we apply the above general method to numerical calculations using Metropolis Monte Carlo simulations of energy expectation values and energy spectra. We analyze and compare the ensuing increase in efficiency of several orders of magnitude.
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
En<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.