12.2: The Born Approximation
- Page ID
- 28684
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.