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,

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