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Suggested Problem Set on finding the points where a function is analytic.
NOTES FOR Session-01 Jul 17, 2022
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.
Does there exist an invertible matrix \(S\) such that \[ S \gamma_\mu S^{-1} = \gamma_\mu'\] where \[\gamma_1'= \gamma_2\gamma_3, \quad \gamma_2'=\gamma_3\gamma_1, \quad \gamma_3'= \gamma_1\gamma_2, \gamma_4'=\gamma_5 \gamma_4?\]
Consider a particle of mass \(\mu\) moving in a potential \[ V(r) = \frac{1}{2}\mu\omega^2 r^2 +\frac{\lambda^2}{2\mu r^2}. \]
- Find condition on energy \(E\) and angular momentum \(L\) for circular orbits to exist.
- Does there exist a circular orbit for \(L=0\)?
- Assume orbital angular momentum \(L=0\), energy \(E= \frac{25}{2}\mu\omega^2a^2\), \(\lambda=12\mu\omega^2\) Use initial conditions \[ r(t)\big|_{t=0} = 4a;\quad \dot{r}(t)\big|_{t=0} = 0 \text{ and } \dot{\theta(t)}\big|_{t=0}=0 \] solve the equations of motion and obtain \(r\), \(\theta\) as function of time. Describe the motion that takes place under conditions specified here.
Find rotation matrix for a rotation by an angle \(\alpha\) about the axis \(1,2,1\) where \(\cos\alpha=\frac{3}{5}, \sin\alpha =\frac{4}{5}\).
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