12.2: The Born Approximation

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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.