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I'm having some trouble understanding this, any help would be appreciated:

Calculate the density of states for a free particle with momentum [tex]\[

\hbar k

\]

[/tex] for the angles between [tex]\[

\left[ {\theta _0 ,\theta _0 + d\theta } \right]

\]

[/tex] relative to an electric field [tex]\[

\vec \varepsilon

\]

[/tex], in the ultra-relative limit [tex]\[

E \cong pc

\]

[/tex].

In my solutions, the first thing they do is to say: Well, as usual, first find [tex]\[

N\left( E \right)

\]

[/tex] and then take the derivative with respect to [tex]\[

E

\]

[/tex], but that's OK. The problem is how they calculate [tex]\[

N\left( E \right)

\]

[/tex]:

[tex]\[

N\left( E \right) = \frac{V}{{h^3 }}\int\limits_{\theta \in \left[ {\theta _0 ,\theta _0 + d\theta } \right]} {d^3 p} = \frac{V}{{h^3 }}2\pi \sin \left( {\theta _0 } \right)d\theta \int\limits_0^{p_{\max } } {p^2 d^2 p}

\]

[/tex]

I can live with the first move. But I don't understand where this sine comes from, or [tex]\[

2\pi

\]

[/tex], and that other integral... help?

Thanks in advance!