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I recently read a question on fermi's golden rule posted here:

Fermi's Golden Rule and Density of States

However, I do not really know how you would go about obtaining a value for the density of states? Classmates are suggesting that it is the number of electrons allowed in an energy level but we are unsure which energy levels we should consider if this were true.

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The density of states is the density of particular things, namely the states. You should better know what "density" means and what "states" mean but if you know the two parts, you simply have to know the combination "density of states", too. It's just the number of mutually orthogonal (energy eigen)states with energy eigenvalue equal to something plus minus a differential, per unit energy. It's $dN/dE$. If you want to calculate the value of it from something else, you must tell us what is the something else. –  Luboš Motl Jan 18 '13 at 14:32

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The density of states $D(E)$ is the number of states at a given energy $E$. For a finite system, this is rather boring, because there are only states at discrete energy levels - that's what you're talking about. Assume a system with the energy levels $E_1$,$E_2$,$E_3$,$E_4$, then the density of states is

$D(E)=\sum_{i=0}^4\delta(E-E_i)$.

This expression says that there is one state at the energies $E_i$, but none elsewhere. For continuous systems, this is more interesting. The number of states can differ between energies, e.g. in Silicon: http://nanohub.org/resource_files/2011/11/12603/slides/026.01.jpg. This is particularly important because the gap in this picture between 0 and approx. 1 eV is what makes silicon a semiconductor.

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