14.1: Fundamentals of Scattering Theory

Consider time-independent, energy conserving scattering in which the Hamiltonian of the system is written \[H = H_0 + V({\bf r}),\] where \[H_0 = \frac{p^{\,2}}{2\,m} \equiv - \frac{\hbar^{\,2}}{2\,m}\,\nabla^{\,2}\] is the Hamiltonian of a free particle of mass \(m\), and \(V({\bf r})\) the scattering potential. This potential is assumed to only be non-zero in a fairly localized region close to the origin. Let \[\psi_0({\bf r}) = \sqrt{n}\,{\rm e}^{\,{\rm i}\,{\bf k}\cdot {\bf r}}\] represent an incident beam of particles, of number density \(n\), and velocity \({\bf v} = \hbar\,{\bf k}/m\). [See Equation ([e14.14gg]).] Of course, \[H_0\,\psi_0= E\,\psi_0,\] where \(E = \hbar^{\,2}\,k^{\,2}/(2\,m)\) is the particle energy. Schrödinger’s equation for the scattering problem is \[(H_0+V)\,\psi = E\,\psi,\] subject to the boundary condition \(\psi\rightarrow\psi_0\) as \(V\rightarrow 0\).

The previous equation can be rearranged to give \[\label{e15.6} (\nabla^{\,2}+k^{\,2})\,\psi = \frac{2\,m}{\hbar^{\,2}}\,V\,\psi.\] Now, \[(\nabla^{\,2}+k^{\,2})\,u({\bf r}) = \rho({\bf r})\] is known as the Helmholtz equation. The solution to this equation is well known \[u({\bf r}) = u_0({\bf r}) - \int \frac{{\rm e}^{\,{\rm i}\,k\,|{\bf r}-{\bf r}'|}} {4\pi\,|{\bf r}-{\bf r}'|}\,\rho({\bf r}')\,d^{\,3}{\bf r}'.\] Here, \(u_0({\bf r})\) is any solution of \((\nabla^{\,2}+k^{\,2})\,u_0 = 0\). Hence, Equation ([e15.6]) can be inverted, subject to the boundary condition \(\psi\rightarrow\psi_0\) as \(V\rightarrow 0\), to give

\begin{equation}\psi(\mathbf{r})=\psi_{0}(\mathbf{r})-\frac{2 m}{\hbar^{2}} \int \frac{\mathbf{e}^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} V\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{r}^{\prime}\right) d^{3} \mathbf{r}^{\prime}\end{equation}

Let us calculate the value of the wavefunction \(\psi({\bf r})\) well outside the scattering region. Now, if \(r\gg r'\) then \[|{\bf r}-{\bf r}'| \simeq r - \hat{\bf r}\cdot {\bf r}'\] to first-order in \(r'/r\), where \(\hat{\bf r}/r\) is a unit vector that points from the scattering region to the observation point. It is helpful to define \({\bf k}'=k\,\hat{\bf r}\). This is the wavevector for particles with the same energy as the incoming particles (i.e., \(k'=k\)) that propagate from the scattering region to the observation point. Equation ([e15.9]) reduces to

\[\label{e15.11} \psi({\bf r}) \simeq \sqrt{n}\left[{\rm e}^{\,{\rm i}\,{\bf k}\cdot{\bf r}} + \frac{e^{\,{\rm i}\,k\,r}}{r}\,f({\bf k}, {\bf k}')\right],\] where

\[\label{e5.12} f({\bf k},{\bf k}') = -\frac{m}{2\pi \sqrt{n}\,\hbar^{\,2}}\int {\rm e}^{-{\rm i}\,{\bf k}'\cdot{\bf r}'}\,V({\bf r}')\,\psi({\bf r}')\,d^{\,3}{\bf r}'.\] The first term on the right-hand side of Equation ([e15.11]) represents the incident particle beam, whereas the second term represents an outgoing spherical wave of scattered particles.

The differential scattering cross-section, \(d\sigma/d{\mit\Omega}\), is defined as the number of particles per unit time scattered into an element of solid angle \(d{\mit\Omega}\), divided by the incident particle flux. From Section [s7.2], the probability flux (i.e., the particle flux) associated with a wavefunction \(\psi\) is \[{\bf j} = \frac{\hbar}{m}\,{\rm Im}(\psi^\ast\,\nabla\psi).\] Thus, the particle flux associated with the incident wavefunction \(\psi_0\) is

\[\label{e14.14gg} {\bf j} = n\,{\bf v},\] where \({\bf v}=\hbar\,{\bf k}/m\) is the velocity of the incident particles. Likewise, the particle flux associated with the scattered wavefunction \(\psi-\psi_0\) is \[{\bf j}' = n\,\frac{|f({\bf k},{\bf k}')|^{\,2}}{r^{\,2}}\,{\bf v}',\] where \({\bf v}' = \hbar\,{\bf k}'/m\) is the velocity of the scattered particles. Now, \[\frac{d\sigma}{d{\mit\Omega}}\,d{\mit\Omega} = \frac{r^{\,2}\,d{\mit\Omega}\,|{\bf j}'|}{|{\bf j}|},\] which yields

\[\label{e15.17} \frac{d\sigma}{d{\mit\Omega}} = |f({\bf k},{\bf k}')|^{\,2}.\] Thus, \(|f({\bf k},{\bf k}')|^{\,2}\) gives the differential cross-section for particles with incident velocity \({\bf v}=\hbar\,{\bf k}/m\) to be scattered such that their final velocities are directed into a range of solid angles \(d{\mit\Omega}\) about \({\bf v}'=\hbar\,{\bf k}'/m\). Note that the scattering conserves energy, so that \(|{\bf v}'|=|{\bf v}|\) and \(|{\bf k}'|=|{\bf k}|\).

Contributors and Attributions

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

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