PX284 - H1 - density of states (DOS)

ψ_{x} = A e^{i k_{x} x}

where, $k_{x}$ is the $x$ -component of the wavevector, $| \vec{k} | = 2 π / λ$

ψ_{x} (L / 2) = ψ_{x} (- L / 2) ⟹ e^{i k_{x} L / 2} = e^{- i k_{x} L / 2}

k_{x} = \frac{2 π}{L} n_{x}

where, $n_{x} \in Z$

\begin{matrix} k_{y} = \frac{2 π}{L} n_{y} \\ k_{z} = \frac{2 π}{L} n_{z} \end{matrix}

thus, the wavevector is quantised into allowed states
considering that the particles are equally distributed along the cube, with a separation of $2 π / L$
in 3D, the volume occupied by one $k$ -state is: $2 π / L \times 2 π / L \times 2 π / L$

density of states

$g (k)$ is defined by $g (k) d k =$ number of allowed states with wavevector whose magnitudes lie between $k$ and $k + d k$

in 3D, this must be given by the volume of spherical shell of thickness, $d k$ , and radius, $k$ , divided by the volume of one $k$ -state:

g (k) d k = 4 π k^{2} \frac{d k}{(2 π / L)^{3}}

g (k) d k = \frac{V k^{2}}{2 π^{2}} d k

where, $V$ is the volume of the container