For a Dirac particle moving in a central potential, show that the orbital angular momentum is not a constant of motion.

For a static (time-independent) perturbation V suddenly switched on at time t=0:

show that the transition probability for a transition of the system from a state 

 up to first order in perturbation theory is just:  How does the transition probability change if the time of application is doubled in the limit t→0. (6+4)

A particle is in the ground state of the simple harmonic oscillator of frequency. A t =0 frequency of the oscillator changes from particle to remain in the ground state at  Calculate the probability for the particle to remain in the ground state at t > 0

Consider the motion of a quantum particle in a potential V(x)-ax. Use the WKB approximation to determine how the energy of the bound state E, varies with n and a for large values of n.

Determine the upper bound to the ground state energy for a particle in a one-dimensional box of length L. using the trial wave function:

Consider the following symmetric two-dimensional infinite potential well  otherwise

Determine the first order perturbation correction to the energy eigenvalue of the two-fold degenerate first excited state, for the following perturbation: 

For a system of two particles each with angular momentum one

(i) Construct the normalized states of highest and second highest J_z for total angular momentum 2.

(i) Construct the normalized state of highest J₂ for total angular momentum 1. 

Calculate the ground state energy for a system of five identical spin  particles placed In a one-dimensional simple harmonic oscillator potential of frequency

Construct the wave function  for a system of 3 identical fermions in the states  respectively of an one-dimensional box of size A 

Show that for a Hamiltonian  to commute with the time reversal operator the potential function V(x) must be real. 

Write the space translation operator in quantum mechanics ) ˆT for an infinitesimal translation (a). Hence evaluate the commutators: and

A charged particle is moving with relativistic speed in a static, uniform electric field E-(E,0,0) along the x-axis. Assume that the particle is initially at rest so that the motion is effectively one-dimensional. Express the energy of the particle as a function of instantaneous position x(t) and solve the resulting equation of motion to obtain the value of x(t).

Explain the phenomenon of time dilation and obtain expression relating the time intervals between two event observed by different inertial observers.

Obtain the expressions for scalar and vector potentials due to an oscillating electric dipole at a point located at large distance from the dipole.

Show that the retarded scalar potential given by

and the retarded vector potential given by

,

satisfy the Lorentz gauge condition.

A plane electromagnetic wave propagating in z-direction is incident on the boundary between two linear dielectric media. The boundary is in the x-y plane perpendicular to the direction of propagation of the wave. Using the boundary conditions on the electric and magnetic fields associated with the wave, show that if the incident wave is plane polarised then the reflected and transmitted waves also have the same polarisation.

Using Fourier integral theorem, obtain the expression for the velocity of a wave packet in a medium for which the dispersion relation is given as 

Using Maxwell's equations, obtain the boundary conditions for D, E, B, and at the interface of two dielectric media.

 At time t= 0, a particle having charge q is placed at rest at the point (0, 0) in a region of crossed uniform static electric and magnetic fields specified by Describe the motion of the particle and show that the motion is periodic.

Obtain the expressions for inhomogeneous Maxwell's equations in terms of scalar and vector potentials.