# 12.3: Box Normalisation and Density of Final States


Plane-wave states have wavefunctions of the form: $$u_{{\bf k},\omega} ({\bf r}) = C \text{ exp}(i({\bf k.r} −\omega t))$$ with $$C$$ a normalisation constant. Because plane-wave states are not properly normalisable we employ the trick of normalising them in a large (relative to potential range) cubic box of side $$L$$ with periodic boundary conditions. We then take the limit $$L \rightarrow \infty$$ at the end of the calculation.

Thus we require that

$\int \int \int_{box} u^*_{{\bf k},\omega} ({\bf r})u_{{\bf k},\omega} ({\bf r}) d\tau = |C|^2 \int \int \int_{box} d\tau = |C|^2 L^3 = 1 \nonumber$

giving for the normalised eigenfunctions: $$u_{{\bf k},\omega} ({\bf r}) = L^{−3/2} \text{ exp}(i{\bf k.r} − \omega t)$$

Of course, enclosing the system in a finite box has the consequence that the allowed momentum eigenvalues are no longer continuous but discrete. With periodic boundary conditions

$u(− \frac{L}{2} , y, z) = u( \frac{L}{2} , y, z), \quad \text{ etc.} \nonumber$

the momentum eigenvalues are forced to be of the form

$p \equiv \hbar {\bf k} = \frac{2\pi \hbar}{L} (n_x, n_y, n_z), \quad \text{ with } n_x, n_y, n_z = 0, \pm 1, \pm 2, ... \nonumber$

For sufficiently large $$L$$, we can approximate the continuous spectrum arbitrarily closely.

Any possible final-state wave-vector, $${\bf k}$$, corresponds to a point in wave-vector space with coordinates $$(k_x, k_y, k_z)$$. The points form a cubic lattice with lattice spacing $$2\pi /L$$. Thus the volume of $$k$$–space per lattice point is $$(2\pi /L)^3$$, and the number of states in a volume element $$d^3{\bf k}$$ is

$\left(\frac{L}{2\pi} \right)^3 d^3{\bf k}' = \left(\frac{L}{2\pi}\right)^3 k^2 dk d\Omega \nonumber$

We require $$g(E_k)$$, the density of states per unit energy, where: $$E_k = \hbar^2 k^2/2m$$ is the energy corresponding to wave-vector $${\bf k}'$$. Now, the wave-vectors in the range $${\bf k}' \rightarrow {\bf k}' + d^3{\bf k}'$$ correspond to the energy range $$E_k \rightarrow E_k + dE_k$$, so that

$g(E_k) dE_k = \left(\frac{L}{2\pi}\right)^3 k^2 dk d\Omega \nonumber$

is the number of states with energy in the desired interval and with wave-vector, $${\bf k}'$$, pointing into the solid angle $$d\Omega$$ about the direction $$(\theta , \phi )$$. Noting that $$dE_k = (\hbar^2 k/m)$$ dk yields the final result for the density of states,

$g(E_k) = \frac{L^3 mk}{8\pi^3\hbar^2} d\Omega \nonumber$

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