14.2: Born Approximation

Equation ([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 ([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 ([e5.12]). This procedure is called the Born approximation .

The Born approximation yields $$f({\bf k},{\bf k}') \simeq \frac{m}{2\pi\,\hbar^{\,2}} \int {\rm e}^{\,{\rm i}\,({\bf k}-{\bf k'})\cdot{\bf r}'}\,V({\bf r'})\,d^{\,3}{\bf r}'.$$ 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, $$f({\bf k}', {\bf k}) \simeq - \frac{m}{2\pi\, \hbar^{\,2}} \int\!\int\!\int \exp(\,{\rm i} \, q \,r'\cos\theta') \, V(r')\,r'^{\,2}\, dr'\,\sin\theta' \,d\theta'\,d\phi',$$ giving $$\label{e17.38} f({\bf k}', {\bf k}) \simeq - \frac{2\,m}{\hbar^{\,2}\,q} \int_0^\infty r' \,V(r') \sin(q \,r') \,dr'.$$ Note that $f({\bf k}', {\bf k})$ is just a function of $q$ for a spherically symmetric potential. It is easily demonstrated that $$\label{e17.39} q \equiv |{\bf k} - {\bf k}'| = 2\, k \,\sin (\theta/2),$$ 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 ,

$$\label{e10.35ee} V(r) = \frac{V_0\,\exp(-\mu \,r)}{\mu \,r},$$ where $V_0$ is a constant, and $1/\mu$ measures the “range” of the potential. It follows from Equation ([e17.38]) that $$f(\theta) = - \frac{2\,m \,V_0}{\hbar^{\,2}\,\mu} \frac{1}{q^{\,2} + \mu^{\,2}},$$ because $$\int_0^\infty \exp(-\mu \,r') \,\sin(q\,r') \, dr' = \frac{q}{q^{\,2}+ \mu^{\,2}}.$$ Thus, in the Born approximation, the differential cross-section for scattering by a Yukawa potential is $$\frac{d\sigma}{d {\mit\Omega}} \simeq \left(\frac{2\,m \,V_0}{ \hbar^{\,2}\,\mu}\right)^2 \frac{1}{[2\,k^{\,2}\, (1-\cos\theta) + \mu^{\,2}]^{\,2}},$$ given that $$q^{\,2} = 4\,k^{\,2}\, \sin^2(\theta/2) = 2\,k^{\,2}\, (1-\cos\theta).$$

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 $$\frac{d\sigma}{d{\mit\Omega}} \simeq \left(\frac{2\,m \,Z\, Z'\, e^{\,2}}{4\pi\,\epsilon_0\,\hbar^{\,2}}\right)^2 \frac{1}{ 16 \,k^{\,4}\, \sin^4( \theta/2)}.$$ Recall that $\hbar\, k$ is equivalent to $|{\bf p}|$, so the previous equation can be rewritten $$\label{e17.46} \frac{d\sigma}{d{\mit\Omega}} \simeq\left(\frac{Z \,Z'\, e^{\,2}}{16\pi\,\epsilon_0\,E}\right)^2 \frac{1}{\sin^4(\theta/2)},$$ where $E= p^{\,2}/2\,m$ is the kinetic energy of the incident particles. Of course, Equation ([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 ([e15.9]), that the condition for $\psi({\bf r}) \simeq \psi_0({\bf r})$ in the vicinity of ${\bf r} = {\bf 0}$ is $$\label{e17.47} \left| \frac{m}{2\pi\, \hbar^{\,2}} \int \frac{ \exp(\,{\rm i}\, k \,r')}{r'} \,V({\bf r}')\,d^{\,3}{\bf r'} \right| \ll 1.$$ 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 $$\frac{2\,m}{\hbar^{\,2}} \frac{|V_0|}{\mu^{\,2}} \ll 1$$ as the condition for the validity of the Born approximation. The condition for the Yukawa potential to develop a bound state is $$\frac{2\,m}{\hbar^{\,2}} \frac{|V_0|} {\mu^{\,2}} \geq 2.7,$$ 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 ([e17.47]) yields $$\frac{2\,m}{\hbar^{\,2}} \frac{|V_0|}{\mu \,k} \ll 1.$$ This inequality becomes progressively easier to satisfy as $k$ increases, implying that the Born approximation is more accurate at high incident particle energies.

Contributors and Attributions

• 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$}}$