A straightforward analytical calculation of the s-like energy spectrum of the hydrogen atom is performed entirely within Feynman's path integral formalism. For this purpose the integral transform W = ∫d3rβK(rβ,0), where K(rβ,0) is the density matrix of the hydrogen atom written as a path integral, is calculated by means of the exact summation of a ``modified'' perturbation expansion (W is expanded as a power series in β with β = 1/kT). Performing this summation is equivalent to solving a problem of moments with infinite moments. For a wide class of potentials the perturbation expansion for W converges faster than the power-series expansion for the exponential function (for the Coulomb potential the convergence rate of both expansions is the same). It is shown how the complete energy spectrum can be obtained by this method. It is also illustrated how the wavefunctions might be obtained by transforming W.
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.