Problem/question

[QUE/ME12006]

Consider a particle of mass \(\mu\) moving in a potential \[ V(r) = \frac{1}{2}\mu\omega^2 r^2 +\frac{\lambda^2}{2\mu r^2}. \] (a) Find condition on energy \(E\) and angular momentum \(L\) for circular orbits to exist.
(b) Does there exist a circular orbit for \(L=0\)?
(c) Assume \(L=0, E= \frac{25}{2}\mu\omega^2a^2\), \(\lambda=12\mu\omega^2\) 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.

KApoor

[QUE/ME-08011]

Find rotation matrix for a rotation by an angle \(\alpha\) about the axis \(1,2,1\) where \(\cos\alpha=\frac{3}{5}, \sin\alpha =\frac{4}{5}\).

[QUE/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.

[QUE/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.