On the Feynman path centroid density for Bose-Einstein and Fermi-Dirac statistics

Abstract

Expressions are derived for the Feynman path centroid density of systems of identical particles, namely, Bosons and Fermions. The resulting expressions are applied to a system of two independent particles in a quartic oscillator and the corresponding centroid densities are computed, plotted and compared to the distinguishable particle case. In regions where the particle centroids are close to each other, the Boson centroid density displays an enhanced amplitude in comparison to the distinguishable case. This behavior is attributed to the attractive correlations in Boson systems. The Fermion centroid density, however, can have the peculiar property of being negative in those regions where the particle centroids are in close proximity. This feature is related to the Pauli exclusion principle. This property of the Fermion centroid density rules out its strict interpretation as a probability distribution. Equilibrium properties such as the canonical partition function and the average position were accurately computed using both densities. The Boson density was also used to compute the position autocorrelation function using the Centroid Molecular Dynamics method and the results are in excellent agreement with those of an exact quantum calculation.