= Solution
\b[<Fermi's golden rule> converts a weak transition <matrix element> and the available density of final states into a transition rate.] For a static perturbation $W$ and a continuum,
$$
\boxed{\Gamma_{i\to{\rm all}}=\frac{2\pi}{\hbar}
\int |\langle f|W|i\rangle|^2\delta(E_f-E_i)\,\mathrm dN_f.}
$$
When the <matrix element> is nearly constant on the energy shell, this becomes $(2\pi/\hbar)|W_{fi}|^2\rho(E_i)$. The energy delta function is to be integrated, rather than interpreted as a finite rate to one exactly resonant discrete level. The observation time must resolve a smooth final-state continuum while remaining short enough that first-order depletion is small. A periodically driven perturbation instead gives the corresponding absorption and emission energy shells.
For elastic scattering from a weak localized potential $W(\mathbf r)$, use box-normalized free states $\langle\mathbf r|\mathbf k\rangle=\mathcal V^{-1/2}e^{i\mathbf k\cdot\mathbf r}$. Then
$$
W_{fi}=\frac1{\mathcal V}\widetilde W(\mathbf q),\qquad
\widetilde W(\mathbf q)=\int e^{-i\mathbf q\cdot\mathbf r}
W(\mathbf r)\,\mathrm d^3r,\qquad
\mathbf q=\mathbf k_f-\mathbf k_i.
$$
The number of final states is $\mathcal V k_f^2\,\mathrm dk_f\,\mathrm d\Omega/(2\pi)^3$. With $E_f=\hbar^2k_f^2/(2m)$, the density per energy and <solid angle> is $\mathcal V m k_f/[(2\pi)^3\hbar^2]$. Dividing the resulting golden-rule rate by the incident single-particle flux $j_i=\hbar k_i/(m\mathcal V)$ gives
$$
\frac{\mathrm d\sigma}{\mathrm d\Omega}
=\frac{m^2}{4\pi^2\hbar^4}\frac{k_f}{k_i}
|\widetilde W(\mathbf q)|^2.
$$
This is the <golden-rule derivation of the Born cross-section>. For the elastic problem $k_f=k_i$, so the first Born <scattering amplitude> and cross-section are
$$
\boxed{f_B(\mathbf k_f,\mathbf k_i)
=-\frac{m}{2\pi\hbar^2}\widetilde W(\mathbf q),\qquad
\frac{\mathrm d\sigma}{\mathrm d\Omega}=|f_B|^2.}
$$
The box volume cancels, as a physical cross-section must. The minus sign is fixed by the outgoing-wave scattering convention; the golden-rule rate alone determines only the modulus.
For example, $W(r)=g e^{-\kappa r}/r$ has $\widetilde W=4\pi g/(q^2+\kappa^2)$, hence $f_B=-2mg/[\hbar^2(q^2+\kappa^2)]$, with $q=2k\sin(\theta/2)$. Replacing the exact scattering state by the incident <plane wave> is the essential <Born approximation>. It can fail for strong scattering, low-energy resonances and situations requiring repeated scattering; a small perturbation and a suitable continuum are separate conditions.<Topics in Quantum Mechanics by David Tong>[§6.3, Born approximation and Yukawa-potential example]{ref}
Back to article page
