- #1
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Definition/Summary
This term most commonly refers to the number of quantum states having energy within a given small energy interval divided by that interval.
Equations
[tex]
g(E)=\sum_{s}\delta(E-E_s)
[/tex]
[tex]
N=\int dE g(E)
[/tex]
The "density of states" need not (but it most often does) refer to states per energy interval. For example, for free particles in a box of volume [itex]\mathcal{V}[/itex], the density of states for a given wavevector [itex]\mathbf{k}[/itex] (rather than energy) is a constant:
[tex]
g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}.
[/tex]
The above equation is the basis for the well-known replacement
[tex]
\sum_{\mathbf{k}}(\ldots)\to\int \mathcal{V}\frac{d^3 k}{{(2\pi)}^3}(\ldots)
[/tex]
Extended explanation
The density of states
[tex]
g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}\;,
[/tex]
results from applying periodic boundry conditions to free waves in a box of volume [itex]\mathcal{V}[/itex] and counting. Thus
[tex]
\delta N = d^3 k g_{\bf k}=d^3 k\frac{\mathcal{V}}{{(2\pi)}^3}\;.
[/tex]
If the energy E only depends on the magnitude of [itex]\mathbf{k}[/itex], E=E(k), then we may also write
[tex]
\delta N = d k k^2 \frac{4\pi \mathcal{V}}{{(2\pi)}^3}
=
\frac{4\pi\mathcal{V}}{{(2\pi)}^3}dE \frac{k^2}{v}\equiv dE g(E)\;,
[/tex]
where
[tex]
v=\frac{dE}{dk}\;,
[/tex]
is the velocity.
For the case where momentum is carried by particles with an effective mass [itex]m^*[/itex] we have
[tex]
k=m^*v\;,
[/tex]
and
[tex]
g(E)=\frac{4\pi \mathcal{V}}{{(2\pi)}^3}m^*\sqrt{2 E m^*}\;.
[/tex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
This term most commonly refers to the number of quantum states having energy within a given small energy interval divided by that interval.
Equations
[tex]
g(E)=\sum_{s}\delta(E-E_s)
[/tex]
[tex]
N=\int dE g(E)
[/tex]
The "density of states" need not (but it most often does) refer to states per energy interval. For example, for free particles in a box of volume [itex]\mathcal{V}[/itex], the density of states for a given wavevector [itex]\mathbf{k}[/itex] (rather than energy) is a constant:
[tex]
g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}.
[/tex]
The above equation is the basis for the well-known replacement
[tex]
\sum_{\mathbf{k}}(\ldots)\to\int \mathcal{V}\frac{d^3 k}{{(2\pi)}^3}(\ldots)
[/tex]
Extended explanation
The density of states
[tex]
g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}\;,
[/tex]
results from applying periodic boundry conditions to free waves in a box of volume [itex]\mathcal{V}[/itex] and counting. Thus
[tex]
\delta N = d^3 k g_{\bf k}=d^3 k\frac{\mathcal{V}}{{(2\pi)}^3}\;.
[/tex]
If the energy E only depends on the magnitude of [itex]\mathbf{k}[/itex], E=E(k), then we may also write
[tex]
\delta N = d k k^2 \frac{4\pi \mathcal{V}}{{(2\pi)}^3}
=
\frac{4\pi\mathcal{V}}{{(2\pi)}^3}dE \frac{k^2}{v}\equiv dE g(E)\;,
[/tex]
where
[tex]
v=\frac{dE}{dk}\;,
[/tex]
is the velocity.
For the case where momentum is carried by particles with an effective mass [itex]m^*[/itex] we have
[tex]
k=m^*v\;,
[/tex]
and
[tex]
g(E)=\frac{4\pi \mathcal{V}}{{(2\pi)}^3}m^*\sqrt{2 E m^*}\;.
[/tex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!