# Fermionic Phonons: Exact Analytic Results and Quantum Statistical Mechanics for a One Dimensional Harmonic Crystal

## TL;DR

This paper derives exact analytic solutions for the energy states of a one-dimensional harmonic crystal and explores their thermal properties, revealing the need for many energy levels for accurate modeling at moderate temperatures.

## Contribution

It provides the first exact analytic expressions for energy eigenvalues and eigenfunctions of a 1D harmonic crystal, along with numerical analysis of thermal properties.

## Key findings

- Large number of energy levels needed for accurate results
- Analytic expressions for eigenvalues and eigenfunctions
- Temperature-dependent energy and density profiles

## Abstract

Analytic expressions for the energy eigenvalues and eigenfunctions of a one-dimensional harmonic crystal are obtained. The average energy and density profiles are obtained numerically as a function of temperature. A surprisingly large number of energy levels (eg.\ 5,000 for 4 particles) are required for reliable results at even moderate temperatures.