14.1: Fundamentals of Scattering Theory

Consider time-independent, energy conserving scattering in which the Hamiltonian of the system is written

\[H = H_0 + V({\bf r}), \nonumber \]

where

\[H_0 = \frac{p^2}{2\,m} \equiv - \frac{\hbar^2}{2\,m}\,\nabla^2 \nonumber \]

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

\[\psi_0({\bf r}) = \sqrt{n}\,\rm e^{\,{\rm i}\,{\bf k}\cdot {\bf r}} \nonumber \]

represent an incident beam of particles, of number density \(n\), and velocity \({\bf v} = \hbar\,{\bf k}/m\). [See Equation \ref{e14.14gg}.] Of course,

\[H_0\,\psi_0= E\,\psi_0, \nonumber \]

where \(E = \hbar^2\,k^2/(2\,m)\) is the particle energy. Schrödinger’s equation for the scattering problem is

\[(H_0+V)\,\psi = E\,\psi, \nonumber \]

subject to the boundary condition \(\psi\rightarrow\psi_0\) as \(V\rightarrow 0\).

The previous equation can be rearranged to give

\[ (\nabla^2+k^2)\,\psi = \frac{2\,m}{\hbar^2}\,V\,\psi. \label{e15.6} \]

Now,

\[(\nabla^2+k^2)\,u({\bf r}) = \rho({\bf r}) \nonumber \]

is known as the Helmholtz equation. The solution to this equation is well known

\[u({\bf r}) = u_0({\bf r}) - \int \frac{\rm e^{\,{\rm i}\,k\,|{\bf r}-{\bf r}'|}} {4\pi\,|{\bf r}-{\bf r}'|}\,\rho({\bf r}')\,d^{\,3}{\bf r}'. \nonumber \]

Here, \(u_0({\bf r})\) is any solution of \((\nabla^2+k^2)\,u_0 = 0\). Hence, Equation \ref{e15.6} can be inverted, subject to the boundary condition \(\psi\rightarrow\psi_0\) as \(V\rightarrow 0\), to give

\[ \psi({\bf r}) = \psi_0({\bf r})- \frac{2\,m}{\hbar^2} \int\frac{\rm e^{\,{\rm i}\,k\,|{\bf r}-{\bf r}'|}} {4\pi\,|{\bf r}-{\bf r}'|}\,V({\bf r}')\,\psi({\bf r}')\,d^{\,3}{\bf r}'. \label{e15.9} \]

Let us calculate the value of the wavefunction \(\psi({\bf r})\) well outside the scattering region. Now, if \(r\gg r'\) then

\[|{\bf r}-{\bf r}'| \simeq r - \hat{\bf r}\cdot {\bf r}' \nonumber \]

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 \ref{e15.9} reduces to

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

where

\[ f({\bf k},{\bf k}') = -\frac{m}{2\pi \sqrt{n}\,\hbar^2}\int \rm e^{-{\rm i}\,{\bf k}'\cdot{\bf r}'}\,V({\bf r}')\,\psi({\bf r}')\,d^{\,3}{\bf r}'. \label{e5.12} \]

The first term on the right-hand side of Equation \ref{e15.11} represents the incident particle beam, whereas the second term represents an outgoing spherical wave of scattered particles.

The differential scattering cross-section, \(d\sigma/d\Omega\), is defined as the number of particles per unit time scattered into an element of solid angle \(d\Omega\), divided by the incident particle flux. From Section [s7.2], the probability flux (i.e., the particle flux) associated with a wavefunction \(\psi\) is

\[{\bf j} = \frac{\hbar}{m}\,{\rm Im}(\psi^\ast\,\nabla\psi). \nonumber \]

Thus, the particle flux associated with the incident wavefunction \(\psi_0\) is

\[ {\bf j} = n\,{\bf v}, \label{e14.14gg} \]

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

\[{\bf j}' = n\,\frac{|f({\bf k},{\bf k}')|^2}{r^2}\,{\bf v}', \nonumber \]

where \({\bf v}' = \hbar\,{\bf k}'/m\) is the velocity of the scattered particles. Now,

\[\frac{d\sigma}{d\Omega}\,d\Omega = \frac{r^2\,d\Omega\,|{\bf j}'|}{|{\bf j}|}, \nonumber \]

which yields

\[ \frac{d\sigma}{d\Omega} = |f({\bf k},{\bf k}')|^2. \label{e15.17} \]

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 \(d\Omega\) about \({\bf v}'=\hbar\,{\bf k}'/m\). Note that the scattering conserves energy, so that \(|{\bf v}'|=|{\bf v}|\) and \(|{\bf k}'|=|{\bf k}|\).