= One-dimensional free-electron density of states
{title2=$g_1(E)=L\sqrt{2m}/(\pi\hbar\sqrt E)$}
For a length $L$ and parabolic free-electron <dispersion relation> $E=\hbar^2k^2/(2m)$, the displayed total <density of states> includes both signs of $k$ and both <spin> states. Per spin it is half as large. Its $E^{-1/2}$ dependence follows by transforming the constant signed-$k$ density $L/(2\pi)$ to <energy> and including both momenta. The continuum singularity at the band bottom is integrable; a finite specimen still has discrete levels.
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