Ph.D.

selection test
Department of Physics
Indian Institute of Technology, Kanpur
December 5, 2017 Time : 9:30 – 11:30 AM Maximum marks : 70

Question 1

(A) Consider a particle in an infinite potential well [the potential V (x) = 0 for 0 < x < L,
otherwise V (x) = ]. The quantum system is described by the energy eigenvalues En and the
corresponding normalized eigenstates 𝜙𝑛 (𝑥) with n = 1, 2, 3,…..

At time t = 0, a particle in the infinite well is in the state given by

1 1 1
𝜓(𝑥, 0) = √3 𝜙1 (𝑥) + √6 𝜙2 (𝑥) + √2 𝜙3 (𝑥) .

(a) Write down the expression for 𝜓(𝑥, 𝑡) [1 mark]

(b) Calculate the expectation value of the energy for the particle described by 𝜓(𝑥, 𝑡). Write your
answer in terms of E1. [3 marks]
(B) Consider a spherically symmetric rigid rotor with moment of inertia Ix = Iy = Iz = Io. Its
Hamiltonian is given by
𝑳2
𝐻=
2𝐼𝑜

with L = r  p is the orbital angular momentum operator.

(a) What are the energy eigenstates and eigenvalues for this quantum rigid rotor? [1 mark]

(b) Now suppose the moment of inertia in the z-direction becomes Iz = (1 + ) Io, where ( << 1)
and with the other two moments unchanged i.e Ix = Iy = Io. What are the new energy eigenstates
and eigenvalues? [5 marks]

Question 2
A neutral spherical ball with radius R and dielectric permittivity  2 is kept inside an infinite
dielectric media with permittivity 1 . The whole system is placed in an electric field which is

uniform far away from the sphere and is given by E  E0 zˆ . After solving the Laplace’s equation
in spherical coordinates, the following solutions are obtained for the potential:

1
 31
V ( r  R)   E0 rcos ,
 2  21

   R3
V ( r  R)   E0 rcos  2 1 E0 cos ,
 221 r 2

where  is the angle the position vector r makes with the direction of the external electric field
and all the other symbols have their usual meaning.

Using the above information,

(a) Find out the electric field inside a spherical cavity of radius R which is hollowed out from an
infinite dielectric media of permittivity ε. The whole system is placed in an electric field which

is uniform far away from the sphere and is given by E  E0 zˆ . Comment on the magnitude and
direction of the electric field with respect to the external field.
[3 marks]
(b) Find out the electric field outside the spherical cavity but inside the dielectric media. [3 marks]
(c) Plot the magnitude of electric field along the z-axis. [2 marks]
(d) Sketch the electric field lines. [2 marks]
Assume isotropic, linear and homogeneous dielectrics.

Question 3

(A) The rate of a particular chemical reaction A + B  C is proportional to the concentrations of

the reactants A and B. Given that C(t = 0) = 0, and

dC(t)/dt =  [A(0) - C(t)] [B(0) - C(t)], where  is a constant.

(a) Find C(t) for A(0)  B(0). [4 marks]

(b) Find C(t) for A(0) = B(0). [3 marks]

(B) Given that m is an integer, and f(z) = zm, calculate the contour integral of f(z) over a unit
circle, with origin at z = 0. [3 marks]

Question 4

(A) A particle of mass m is constrained to move on a curve in the vertical plane defined by the
parametric equation: x = l(2+ sin2); y = l(1 - cos2). There is the usual constant
gravitational force acting in the vertical y direction.

2
 (a) Calculate the Hamiltonian of the system. Is the Hamiltonian conserved? Is the energy of
the system conserved? For each case give proper justification to your answer. [3 marks]
(b) Calculate the action integral for the system. [4 marks]

(B) Three equal mass points (mass 10 g) are located at (a, 0, 0); (0, a, 2a); and (0, 2a, a). Obtain
the principal moments of inertia of the system. Take a = 2 cm. [3 marks]

Question 5

(A) A digital stopwatch can read at a precision of 1/10 of a second. However, the display of the
watch is damaged and the tens’ place of second is not readable (the display looks like:
00:00:X0.0). Where “X” represents the tens place of a second which is not readable. What is the
effective measurement precision of this digital stopwatch? Explain your answer briefly.
[2 marks]
(B) Random measurement uncertainties are inevitably introduced in any measurement and are
propagated to the processed data. The time period (T) of a pendulum is measured in two
different ways. In one experiment the total time for 50 oscillations (T50) is measured and the
time period is calculated as T = (T50 / 50). In another experiment, time for each complete
oscillation (T1) is measured 50 times and the time period is calculated by taking mean, i.e. T =
(<T1>50). Compare the propagated uncertainties in these two cases and thus conclude which
between the two, statistically, gives more accurate value for the time period? [3 marks]
(C) When a light beam of intensity I0 passes through a neutral density (ND) filter, the intensity of
the transmitted light (It) gets reduced by a factor 10- i.e. It = I0 10-, where  is the optical
density of the filter. In an experiment a rectangular ND filter (length = 2l) is used, where 
changes linearly from a maximum value of d at the center to 0 at both ends (± l) along its length
(see figure below). A laser beam is passed through the middle of this ND filter. Now, if the
ND filter starts performing simple harmonic motion along the length with time period T and
amplitude l. Derive the transmitted intensity of the laser beam as a function of time. What is
the time period of oscillation in the transmitted intensity? Does it oscillate in a simple harmonic
manner? What is the minimum time that it needs to be averaged over to calculate the time
averaged transmitted intensity? [5 marks]

3
Question 6

(A) Find the Thevenin equivalent circuit (across RL) for the following network:
[5 marks]

(B) Draw the circuit diagram for negative feedback amplifiers of following specifications using
an ideal Op-Amp (IC-741). Each circuit must contain three (and only three) 10 kΩ resistors.
[5 marks]

(a) Av(CL) = -2 and RI = 10 kΩ.

(b) Av(CL) = -2 and RI = 5 kΩ.
(c) Av(CL) = -0.5 and RI = 10 kΩ.
(d) Av(CL) = +3
(e) Av(CL) = +3 and RF = 10 kΩ.

Here, Av(CL) is the closed loop gain. RI is the input resistor and RF is the feedback resistor.

Question 7

Consider a system of six distinguishable, non-interacting spins. Each spin can only occupy two
states: `up' and `down'. For the first five spins, the energy levels are - for an up spin and + for
down spin. However, the sixth spin has twice the magnetic moment and, therefore, it's energy
levels are -2 and +2. If the total energy is -3, calculate (a) the entropy and (b) the average
number of up spins. [7marks + 3 marks]

4
Useful formulae (In spherical coordinates):