Calculating and visualizing the density of states for simple quantum mechanical systems

TL;DR

This paper introduces a graphical method for calculating and visualizing the density of states in simple quantum systems, emphasizing the influence of system dimensions and extending to multi-particle bosonic systems.

Contribution

It presents a straightforward computational approach to relate quantum energy levels with density of states and applies it to various geometries and multi-particle systems.

Findings

Density of states depends primarily on the system's dimension.

Graphical approach effectively visualizes degeneracy and cumulative states.

Multi-particle density of states can be derived from single-particle spectra.

Abstract

We present a graphical approach to understanding the degeneracy, density of states, and cumulative state number for some simple quantum systems. By taking advantage of basic computing operations we define a straightforward procedure for determining the relationship between discrete quantum energy levels and the corresponding density of states and cumulative level number. The density of states for a particle in a rigid box of various shapes and dimensions is examined and graphed. It is seen that the dimension of the box, rather than its shape, is the most important feature. In addition, we look at the density of states for a multi-particle system of identical bosons built on the single-particle spectra of those boxes. A simple model is used to explain how the $N$-particle density of states arises from the single particle system it is based on.