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

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