Developments in the Centroid Phase Space Formulation of Quantum Statistical Mechanics

Abstract

The centroid formalism provides a phase space representation of quantum statistical mechanics based on the Feynman path integral. Real time quantum correlation functions can be exactly calculated using the centroid formalism, though this requires diagonalizing the system Hamiltonian which is intractable for large collections of molecules. A computational method for computing real time correlation functions called centroid molecular dynamics (CMD) has been formulated to circumvent this issue though the results are approximations. The centroid formalism had previously only been able to treat systems moving in Euclidean space. This is insufficient to capture rotational motion and intramolecular torsions, which may be viewed as motion in a constrained subspace of the Euclidean space. Herein we present a method for incorporating this type of motion into the centroid formalism and test the validity by examining the motion of a particle on a ring. Past work has also seen the centroid formalism extended to pairs of particles obeying Bose-Einstein and Fermi-Dirac statistics by way of a projection operator. In this work we examine the case where this projection operator projects onto an individual quantum state. This will allow the centroid formalism, and hence CMD, to be extended to microcanonical ensembles. Results are shown for the quantum harmonic oscillator, quartic well system and double well system.