Quantum canonical ensemble: a projection operator approach

TL;DR

This paper introduces a projection operator method to compute the quantum canonical ensemble's partition function and related thermodynamic quantities for non-interacting particles, overcoming previous computational challenges.

Contribution

The novel projection operator approach allows for constraint-free calculations of the canonical partition function and correlation functions in quantum systems.

Findings

Provides closed-form expressions for $Z_N$ and $F_N$

Applicable to bosonic and fermionic systems in arbitrary dimensions

Demonstrates the method on a 2D fermion gas with N=2 to 500

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.