Quantum canonical ensemble: a projection operator approach

Abstract

Fixing the number of particles $N$, the quantum canonical ensemble imposes a constraint on the occupation numbers of single-particle states. The constraint particularly hampers the systematic calculation of the partition function and any relevant thermodynamic expectation value for arbitrary $N$ since, unlike the case of the grand-canonical ensemble, traces in the $N$-particle Hilbert space fail to factorize into simple traces over single-particle states. In this paper we introduce a projection operator that enables a constraint-free computation of the partition function and its derived quantities, at the price of an angular or contour integration. Being applicable to both bosonic and fermionic non-interacting systems in arbitrary dimensions, the projection operator approach provides closed-form expressions for the partition function $Z_N$ and the Helmholtz free energy $F_{\! N}$ as well as for two- and four-point correlation functions. While appearing only as a secondary quantity in the present context, the chemical potential potential emerges as a by-product from the relation $\mu_N = F_{\! N+1} - F_{\! N}$, as illustrated for a two-dimensional fermion gas with $N$ ranging between 2 and 500.