Find the central force if the orbit of a particle is given by \[ r=a(1+\cos\theta). \]
$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$Write Dirac equation in external electromagentic potential \(A_\mu(x)\). Assume the potentials for Coulomb interactions with a nucleus of charge \(Ze\) to be \[ \vec{A}=0, \quad A_0 = \frac{Ze}{4\pi |x|}. \] Show that the differential cross section for electron scattering from nucleus is given by \[\dd[\sigma]{\Omega}= \left(\dd[\sigma]{\Omega}\right)_\text{R} \Big(1-v^2\sin^2(\theta/2)\Big)\] where \[\left(\dd[\sigma]{\Omega}\right)_\text{R} = \frac{Ze^2}{64\pi m^2v^4 \sin^4(\theta/2)}\] is the Rutherford cross section for nonrelativistic Coulomb scattering
Let \(a_{\mu\nu} \) give the Lorentz transformation \[x'^{\mu}=a_{\mu\nu} x^\nu.\] Show that there exists an invertible matrix \(S\) such that \[S\gamma_\mu S^{-1}= a_{\mu\nu}\gamma^\nu . \]
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