Path integrals for the quantum microcanonical ensemble
Path integral representations for the quantum microcanonical ensemble are presented. In the quantum microcanonical ensemble, two operators are of primary interest. First, rhoinsertion mark=delta(E-Hinsertion mark) corresponds to the microcanonical density matrix and can be used to calculate expectation values. Second, Ninsertion mark=Theta(E-Hinsertion mark) can give the number of states with energy E(n)<E. We consider position matrix elements of both of these operators Omega(x,x('),E)=<x(')|delta(E-Hinsertion mark)|x> and Theta(x,x('),E)=<x(')|straight theta(E-Hinsertion mark)|x>. A path integral formalism leads to exact integral representations for Omega(x,x('),E) and Theta(x,x('),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.
Lawson
JW
Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-1907, USA.