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

# Problem/question

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.

- Write the Lagrangian for free Schrodinger field and obtain an expression for the Hamiltonian.
- 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?

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 \[\gamma_1'= \gamma_2\gamma_3, \quad \gamma_2'=\gamma_3\gamma_1, \quad \gamma_3'= \gamma_1\gamma_2, \gamma_4'=\gamma_5 \gamma_4?\]

Usage Context : Quiz, Comprehension Check, Diagnostic Assessment