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SM-Exm-2017

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

MP-Exm-2017

Problem/QFT/15013

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.


Problem/QFT-01

  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.

Problem/QFT/06007

 

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
\[\bar{u}(p') \gamma_{\alpha_1} \gamma_{\alpha_2}\ldots \gamma_{\alpha_n}
u(p) \to  \bar{u}(p)\gamma_{\alpha_n}\gamma_{\alpha_{n-1}}\ldots
\gamma_{\alpha_1} u(p')) \]

Prob/QFT/06004

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

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