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9.3: Born Approximation

  • Page ID
    1240
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Equation \ref{938} is not particularly useful, as it stands, because the quantity \( \vert\psi\rangle\) . Recall that \( \psi({\bf x})=\langle {\bf x}\vert\psi\rangle\) is the solution of the integral equation

    $ \psi({\bf x}) = \phi({\bf x})-\frac{m}{2\pi\,\hbar^2} \frac{\exp(...
...x'\, \exp(- {\rm i} \,{\bf k}' \cdot {\bf x}')\, V({\bf x}')\, \psi ({\bf x}'),$ \ref{939}

    where \( V\) , as well as the local value of the wavefunction, \( \psi({\bf x})\) , does not differ substantially from the incident wavefunction, \( f({\bf k}', {\bf k})\) by making the substitution

    \( f({\bf k}', {\bf k}) \simeq - \frac{m}{2\pi\, \hbar^2} \int d^3 x'\,\exp\left[\, {\rm i}\, ({\bf k} - {\bf k}')\cdot {\bf x}'\right] V({\bf x}').\) \ref{941}

    Thus, \( V({\bf x})\) with respect to the wavevector \( {\bf q} \equiv {\bf k} - {\bf k}'\) .

    For a spherically symmetric potential,

    $ f({\bf k}', {\bf k}) \simeq - \frac{m}{2\pi\, \hbar^2} \int_0^\in...
...,d\phi'\,r'^{\,2}\,\sin\theta' \,\exp(\,{\rm i} \, q \,r'\cos\theta') \, V(r'),$ \ref{942}

    giving

    \( f({\bf k}', {\bf k})\) is just a function of \( q \equiv \vert{\bf k} - {\bf k}'\vert = 2\, k \,\sin (\theta/2),\) \ref{944}

    where \( {\bf k}\) and \( {\bf k}'\) . In other words, \( {\bf k}\) and \( {\bf k}'\) have the same length, as a consequence of energy conservation.

    Consider scattering by a Yukawa potential

    \( V_0\) is a constant, and \( f(\theta) = - \frac{2\,m \,V_0}{\hbar^2\,\mu} \frac{1}{q^2 + \mu^2},\) \ref{946}

    because

    \( \frac{d\sigma}{d {\mit\Omega}} \simeq \left(\frac{2\,m \,V_0}{ \hbar^2\,\mu}\right)^2 \frac{1}{[4\,k^2\,\sin^2(\theta/2) + \mu^2]^{\,2}}.\) \ref{948}

    The Yukawa potential reduces to the familiar Coulomb potential as \( \mu \rightarrow 0\) , provided that $ V_0/\mu \rightarrow
Z\,Z'\, e^2 / 4\pi\,\epsilon_0$ . In this limit, the Born differential cross-section becomes

    \( \hbar\, k\) is equivalent to \( \frac{d\sigma}{d{\mit\Omega}} \simeq\left(\frac{Z \,Z'\, e^2}{16\pi\,\epsilon_0\,E}\right)^2 \frac{1}{\sin^4(\theta/2)},\) \ref{950}

    where \( \psi({\bf x})\) is not too different from \( \phi({\bf x})\) in the scattering region. It follows, from Equation \ref{922}, that the condition for $ \psi({\bf x})
\simeq \phi({\bf x})$ in the vicinity of \( \left\vert \frac{m}{2\pi\, \hbar^2} \int d^3 x'\,\frac{ \exp(\,{\rm i}\, k \,r')}{r'} \,V({\bf x}') \right\vert \ll 1.\) \ref{951}

    Consider the special case of the Yukawa potential. At low energies, (i.e., \( \exp(\,{\rm i}\,k\, r')\) by unity, giving

    \( \frac{2\,m}{\hbar^2} \frac{\vert V_0\vert} {\mu^2} \geq 2.7,\) \ref{953}

    where \( k\) limit, Equation \ref{951} yields

    \( \frac{2\,m}{\hbar^2} \frac{\vert V_0\vert}{\mu \,k} \ll 1.\) \ref{954}

    This inequality becomes progressively easier to satisfy as \(k\) increases, implying that the Born approximation is more accurate at high incident particle energies.

    Contributors

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)


    This page titled 9.3: Born Approximation is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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