12.3: Box Normalisation and Density of Final States
- Page ID
- 28685
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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,
\[g(E_k) = \frac{L^3 mk}{8\pi^3\hbar^2} d\Omega \nonumber\]