Quantum Statistics and Thermodynamics in the Harmonic Approximation

TL;DR

This paper introduces a method to calculate thermodynamic properties of identical bosons and fermions in a confining potential using harmonic approximation, efficiently separating center of mass and relative motions.

Contribution

It presents a novel approach to compute thermodynamic quantities in the harmonic approximation considering particle symmetry and energy degeneracies, applicable to realistic models.

Findings

Method accurately computes thermodynamic quantities for 2D systems.

Numerical results agree with brute force calculations, demonstrating efficiency.

Applicable to systems with harmonic interactions and realistic parameters.

Abstract

We describe a method to compute thermodynamic quantities in the harmonic approximation for identical bosons and fermions in an external confining field. We use the canonical partition function where only energies and their degeneracies enter. The number of states of given energy and symmetry is found by separating the center of mass motion, and counting the remaining states of given symmetry and excitation energy of the relative motion. The oscillator frequencies that enter the harmonic Hamiltonian can be derived from realistic model parameters and the method corresponds to an effective interaction approach based on harmonic interactions. To demonstrate the method, we apply it to systems in two dimensions. Numerical calculations are compared to a brute force method that is considerably more computationally intensive.