14.2: Born Approximation

Equation \ref{e15.17} is not particularly useful, as it stands, because the quantity \(f({\bf k},{\bf k}')\) depends on the, as yet, unknown wavefunction \(\psi({\bf r})\). [See Equation \ref{e5.12}.] Suppose, however, that the scattering is not particularly strong. In this case, it is reasonable to suppose that the total wavefunction, \(\psi({\bf r})\), does not differ substantially from the incident wavefunction, \(\psi_0({\bf r})\). Thus, we can obtain an expression for \(f({\bf k},{\bf k}')\) by making the substitution \(\psi({\bf r})\rightarrow\psi_0({\bf r}) = \sqrt{n}\,\exp(\,{\rm i}\, {\bf k}\cdot{\bf r})\) in Equation \ref{e5.12}. This procedure is called the Born approximation .

The Born approximation yields

\[f({\bf k},{\bf k}') \simeq \frac{m}{2\pi\,\hbar^2} \int \rm e^{\,{\rm i}\,({\bf k}-{\bf k'})\cdot{\bf r}'}\,V({\bf r'})\,d^{\,3}{\bf r}'. \nonumber \]

Thus, \(f({\bf k},{\bf k}')\) becomes proportional to the Fourier transform of the scattering potential \(V({\bf r})\) with respect to the wavevector \({\bf q} = {\bf k}-{\bf k}'\).

For a spherically symmetric potential,

\[f({\bf k}', {\bf k}) \simeq - \frac{m}{2\pi\, \hbar^2} \int\!\int\!\int \exp(\,{\rm i} \, q \,r'\cos\theta') \, V(r')\,r'^2\, dr'\,\sin\theta' \,d\theta'\,d\phi', \nonumber \]

giving

\[ f({\bf k}', {\bf k}) \simeq - \frac{2\,m}{\hbar^2\,q} \int_0^\infty r' \,V(r') \sin(q \,r') \,dr'. \label{e17.38} \]

Note that \(f({\bf k}', {\bf k})\) is just a function of \(q\) for a spherically symmetric potential. It is easily demonstrated that

\[ q \equiv |{\bf k} - {\bf k}'| = 2\, k \,\sin (\theta/2), \label{e17.39} \]

where \(\theta\) is the angle subtended between the vectors \({\bf k}\) and \({\bf k}'\). In other words, \(\theta\) is the scattering angle. Recall that the vectors \({\bf k}\) and \({\bf k}'\) have the same length, via energy conservation.

Consider scattering by a Yukawa potential ,

\[ V(r) = \frac{V_0\,\exp(-\mu \,r)}{\mu \,r}, \label{e10.35ee} \]

where \(V_0\) is a constant, and \(1/\mu\) measures the “range” of the potential. It follows from Equation \ref{e17.38} that

\[f(\theta) = - \frac{2\,m \,V_0}{\hbar^2\,\mu} \frac{1}{q^2 + \mu^2}, \nonumber \]

because

\[\int_0^\infty \exp(-\mu \,r') \,\sin(q\,r') \, dr' = \frac{q}{q^2+ \mu^2}. \nonumber \]

Thus, in the Born approximation, the differential cross-section for scattering by a Yukawa potential is

\[\frac{d\sigma}{d \Omega} \simeq \left(\frac{2\,m \,V_0}{ \hbar^2\,\mu}\right)^2 \frac{1}{[2\,k^2\, (1-\cos\theta) + \mu^2]^2}, \nonumber \]

given that

\[q^2 = 4\,k^2\, \sin^2(\theta/2) = 2\,k^2\, (1-\cos\theta). \nonumber \]

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

\[\frac{d\sigma}{d\Omega} \simeq \left(\frac{2\,m \,Z\, Z'\, e^2}{4\pi\,\epsilon_0\,\hbar^2}\right)^2 \frac{1}{ 16 \,k^{\,4}\, \sin^4( \theta/2)}. \nonumber \]

Recall that \(\hbar\, k\) is equivalent to \(|{\bf p}|\), so the previous equation can be rewritten

\[ \frac{d\sigma}{d\Omega} \simeq\left(\frac{Z \,Z'\, e^2}{16\pi\,\epsilon_0\,E}\right)^2 \frac{1}{\sin^4(\theta/2)}, \label{e17.46} \]

where \(E= p^2/2\,m\) is the kinetic energy of the incident particles. Of course, Equation \ref{e17.46} is identical to the famous Rutherford scattering cross-section formula of classical physics .

The Born approximation is valid provided that \(\psi({\bf r})\) is not too different from \(\psi_0({\bf r})\) in the scattering region. It follows, from Equation \ref{e15.9}, that the condition for \(\psi({\bf r}) \simeq \psi_0({\bf r})\) in the vicinity of \({\bf r} = {\bf 0}\) is

\[ \left| \frac{m}{2\pi\, \hbar^2} \int \frac{ \exp(\,{\rm i}\, k \,r')}{r'} \,V({\bf r}')\,d^{\,3}{\bf r'} \right| \ll 1. \label{e17.47} \]

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

\[\frac{2\,m}{\hbar^2} \frac{|V_0|}{\mu^2} \ll 1 \nonumber \]

as the condition for the validity of the Born approximation. The condition for the Yukawa potential to develop a bound state is

\[\frac{2\,m}{\hbar^2} \frac{|V_0|} {\mu^2} \geq 2.7, \nonumber \]

where \(V_0\) is negative . Thus, if the potential is strong enough to form a bound state then the Born approximation is likely to break down. In the high-\(k\) limit, Equation \ref{e17.47} yields

\[\frac{2\,m}{\hbar^2} \frac{|V_0|}{\mu \,k} \ll 1. \nonumber \]

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