When energy levels are numerous or closely spaced, it is useful to describe how many states lie in an energy interval rather than list each state individually. The density of states $g(E)$ is defined so that $$g(E),dE$$ is approximately the number of available one-particle states with energies between $E$ and $E+dE$.

Thermal sums can then be approximated by integrals: $$\sum_i f(E_i)\approx\int g(E)f(E),dE.$$ This separates two ingredients: the geometry and dynamics that determine how many states exist at each energy, and the statistics that determine how strongly those states are occupied.

For a free nonrelativistic particle in three dimensions, $E=p^2/(2m)$. Momentum states fill three-dimensional momentum space, so the number of states below momentum $p$ grows as $p^3$. Since $p\propto E^{1/2}$, the cumulative number of states grows as $E^{3/2}$, giving $$g(E)\propto E^{1/2}.$$ The proportionality constant depends on volume, mass and the state-counting convention.

The density of states is central in solid-state and statistical physics. Heat capacities, electron populations, photon spectra and transition rates depend not only on whether an energy is allowed but on how many states are available near that energy. Sharp features in $g(E)$ can therefore create strong observable effects even when the occupation law itself is smooth.