9.3: Born Approximation
- Page ID
- 1240
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
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
Thus, \( V({\bf x})\) with respect to the wavevector \( {\bf q} \equiv {\bf k} - {\bf k}'\) .
For a spherically symmetric potential,
giving
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
because
The Yukawa potential reduces to the familiar Coulomb potential as \( \mu \rightarrow 0\) , provided that . In this limit, the Born differential cross-section becomes
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 in the vicinity of \( \left\vert \frac{m}{2\pi\, \hbar^2} \int d^3 x'\,\frac{ \exp(\,{\rm i}\, k \,r')}{r'} \,V({\bf x}') \right\vert \ll 1.\) Consider the special case of the Yukawa potential. At low energies, (i.e., \( \exp(\,{\rm i}\,k\, r')\) by unity, giving where \( k\) limit, Equation \ref{951} yields This inequality becomes progressively easier to satisfy as \(k\) increases, implying that the Born approximation is more accurate at high incident particle energies. Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)\ref{951} \( \frac{2\,m}{\hbar^2} \frac{\vert V_0\vert} {\mu^2} \geq 2.7,\) \ref{953} \( \frac{2\,m}{\hbar^2} \frac{\vert V_0\vert}{\mu \,k} \ll 1.\) \ref{954} Contributors