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QS 12: Cross sections for non-relativistic potential scattering

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\newcommand{\intp}{\int \frac{{\rm d}^3p}{2p^0}}
\newcommand{\intpp}{\int \frac{{\rm d}^3p'}{2{p'}^0}}
\newcommand{\intx}{\int{\rm d}^3{\rm r}}
\newcommand{\tp}{\otimes}
\newcommand{\tpp}{\tp\cdots\tp}
\newcommand{\kk}[1]{|#1\rangle}
\newcommand{\bb}[1]{\langle #1}
\newcommand{\dd}[1]{\delta_{#1}}
\newcommand{\ddd}[1]{\delta^3(#1)}
\newcommand{\vv}[1]{{\bf #1}}
\newcommand{\molp}{\Omega^{(+)}}
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A beam of particles gets scattered by a fixed immovable target sitting at the origin and represented by a potential $V$. In this case total energy is conserved but momentum is not.

We consider the beam of N spin-less particles with initial state  representing particles with sharp momentum around $\vv{k}$ and such that these particles pass near the origin where the scatterer is located.

We want to find out the number of scattered particles moving with momenta in a narrow range $\Delta$ around $\vv{k'}$.

This number is given by the formula given in the previous section, with the projection operator $\kk{\xi(t)}\bb{\xi(t)}|$ ocurring in $B(t)={\molp}^\dagger\kk{\xi(t)}\bb{\xi(t)}|V\molp$ replaced by
\begin{eqnarray*} \kk{\xi(t)}\bb{\xi(t)}|=\int_{\Delta}{\rm d}^3\vv{k'}
\kk{\vv{k'}}\bb{\vv{k'}}| \end{eqnarray*}

Therefore, we must calculate $\bb{\vv{k}}|B\kk{\vv{k}}$.
\begin{eqnarray*} \bb{\vv{k}}|B\kk{\vv{k}}=\int_{\Delta}{\rm d}^3\vv{k'}
\bb{\vv{k}}|{\molp}^\dagger\kk{\vv{k'}}\bb{\vv{k'}}|V\molp\kk{\vv{k}} \end{eqnarray*}

From the Lippmann-Schwinger equation in energy eigenstates basis, we get
\begin{eqnarray*} \bb{\vv{k'}}|\molp\kk{\vv{k}}=\frac{\bb{\vv{k'}}|V\molp\kk{\vv{k}}}
{E_{\vv{k}}-E_{\vv{k'}}+i\epsilon} \end{eqnarray*}
the term corresponding to identity being zero as $\bb{\vv{k'}}\kk{\vv{k}}=0$.

Substituting the complex conjugate of the matrix element of $\molp$ in the previous formula we get,
\begin{eqnarray*} \bb{\vv{k}}|B\kk{\vv{k}}=\int_{\Delta}{\rm d}^3\vv{k'}
\frac{|T(\vv{k'},\vv{k})|^2}{E_{\vv{k}}-E_{\vv{k'}}+i\epsilon} \end{eqnarray*}
where the transition amplitude $T$ is 
\begin{eqnarray*} T(\vv{k'},\vv{k})\equiv\bb{\vv{k'}}|V\molp\kk{\vv{k}}\end{eqnarray*}
Using the well known formula
\begin{eqnarray}\frac{1}{x-i\epsilon}=P\left(\frac{1}{x}\right) +i\pi\delta(x) \end{eqnarray}
we get
\begin{eqnarray*} {\rm Im}\bb{\vv{k}}|B\kk{\vv{k}}=\pi\int_{\Delta}{\rm d}^3\vv{k'}
|T(\vv{k'},\vv{k})|^2\delta(E_{\vv{k}}-E_{\vv{k'}}) \end{eqnarray*}

We can further simplify the formula by writing  ${\rm d}^3\vv{k'}=m|\vv{k'}|dE_{\vv{k'}}d\Omega_{\vv{k'}}$ and integerating over the Dirac delta function. Therefore
\begin{eqnarray*} n_\xi=(2\pi)^4\hbar^2\rho m|\vv{k}|\int_\Delta d\Omega_{\vv{k'}}
|T(\vv{k'},\vv{k})|^2 \end{eqnarray*}
where it is to be noted that in the transition amplitude now $|\vv{k}|=|\vv{k'}|$.

The number of particles scattering per unit time into the required final states is proportional to the {\bf flux} given by the product of density $\rho$ and velocity of particles $|\vv{k}|/m$. The ratio of $n_\xi$ to flux is a quantity of the dimensions of area called {\bf cross-section}.

The cross section is \begin{eqnarray}\sigma=(2\pi)^4\hbar^2 m^2\int_\Delta d\Omega_{\vv{k'}}
|T(\vv{k'},\vv{k})|^2 \end{eqnarray}

If the range $\Delta$ is very narrow then we have the formula for the {\bf differential cross-section}
\begin{eqnarray}d\sigma=(2\pi)^4\hbar^2 m^2|T(\vv{k'},\vv{k})|^2d\Omega_{\vv{k'}}
\end{eqnarray}

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