12.2: The Born Approximation

We can use time-dependent perturbation theory to do an approximate calculation of the crosssection. Provided that the interaction between particle and scattering center is localized to the region around $r = 0$, we can regard the incident and scattered particles as free when they are far from the scattering center. We just need the result that we obtained for a constant perturbation, Fermi’s Golden Rule, to compute the rate of transitions between the initial state (free particle of momentum ${\bf p}$) to the final state (free particle of momentum ${\bf p'}$).

The Hamiltonian for a single particle being scattered by a fixed potential as

$$\hat{H} = \hat{H}_0 + \hat{V} ({\bf r}) \quad \text{ where } \quad \hat{H}_0 = \frac{{\bf \hat{p}}^2}{2m}, \quad \text{ the kinetic energy operator} \nonumber$$

and treat the potential energy operator, $\hat{V} ({\bf r})$, as the perturbation inducing transitions between the eigenstates of $\hat{H}_0$, which are plane waves.

If we label the initial and final plane-wave states $\Phi_{in} = \text{ exp}(i{\bf k.r}−i\omega t)$ and $\Phi_{scat} = \text{ exp}(i{\bf k'.r}−i\omega' t)$ by their respective wave-vectors, then Fermi’s Golden Rule for the rate of transitions is

$$R = \frac{2\pi}{\hbar} |\langle {\bf k}' |\hat{V} |{\bf k} \rangle|^2 g(E_k) \nonumber$$

where $g(E_k)$ is the density of final states; $g(E_k)dE_k$ is the number of final states with energy in the range $E_k \rightarrow E_k + dE_k$.

The quantity $\langle {\bf k}' |\hat{V} |{\bf k} \rangle$ is known as the matrix element of the perturbation and is usually abbreviated thus

$$V_{{\bf k'k}} \equiv \langle {\bf k}' |\hat{V} |{\bf k} \rangle = \int \int \int u^*_{{\bf k'}} ({\bf r}) V ({\bf r})u_{{\bf k}}({\bf r}) d\tau. \nonumber$$

The time variation has been suppressed here. For constant potential, the only non-zero terms come from $\omega = \omega'$: elastic scattering. For a time oscillating potential (e.g. $V ({\bf r}) \sin \omega_0 t)$ the non-zero contribution comes from $\omega = \omega' \pm \omega_0$: inelastic scattering where the scattered particle gains/loses a quantum of energy from/to the system providing the potential.