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## MP-Cross Roads :: All Areas

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## SM-Exm-2017

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

## 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
\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)}.
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?

## 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  $\gamma_1'= \gamma_2\gamma_3, \quad \gamma_2'=\gamma_3\gamma_1, \quad \gamma_3'= \gamma_1\gamma_2, \gamma_4'=\gamma_5 \gamma_4?$