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Level :M.Sc.             End Semester Examination              Duration : 3hrs



Show that the Rutherford scattering cross section for a second quantized Dirac  particle in an external Coulomb field \((Ze^2/r)\) is given by  \begin{equation}
 \frac{d\sigma}{d\Omega} = \frac{Z^2\alpha^2(1-v^2\sin^2(\theta/2))}{4|\vec{k}|^2 v^2 \sin^4(\theta/2)}.\end{equation}
where \(\vec{k}\) is the momentum of the incident particle and \(\theta \) is the angle of scattering.


  1. Write the Lagrangian for free Schrodinger field and obtain an expression for the Hamiltonian.
  2. Using the Poisson bracket form of equations of motion show that the Galilean boost \[\int d^3 x\psi^\dagger (m~x+ it \hbar \nabla)\psi,\] is a conserved quantity. How do you interpret this conservation law?

Comprehension Check/QM-06

Problem/QFT/06008 Dirac particle in uniform magnetic field

Consider an electron in a uniform and constant magentic field \(\vec{B}\) along the \(z-\)axis. Obtain the most general four component positive energy eigennfunctions. Show that the energy eigenvalues are given by
\[ E= \sqrt{m^2c^4 + c^2p_3^2 + 2ne\hbar c|\vec{B}|}\]
with \(n=0,1,2,...\). List all the constants of motion.



Show that under time reversal \(\bar{u}(p) \to  u(—p)B\) and \(v(p) \to \(\bar{v}(—p) B\). Use this
to show that under \(P T\) together
\[\bar{u}(p') \gamma_{\alpha_1} \gamma_{\alpha_2}\ldots \gamma_{\alpha_n}
u(p) \to  \bar{u}(p)\gamma_{\alpha_n}\gamma_{\alpha_{n-1}}\ldots
\gamma_{\alpha_1} u(p')) \]


Does there exist an invertible matrix \(S\) such that \[ S \gamma_\mu  S^{-1} = \gamma_\mu'\] where  \[\gamma_1'= \gamma_2\gamma_3, \quad \gamma_2'=\gamma_3\gamma_1, \quad \gamma_3'= \gamma_1\gamma_2, \gamma_4'=\gamma_5 \gamma_4?\] 

CCQ/QM-16 Spherically Symmetric Potentials

CCQ/QM-23 Scattering in TIme Dependent Formalism of Quantum Mechanics

Usage Context :  Quiz, Comprehension Check, Diagnostic Assessment