CMI Entrance Exam 2016 Question Paper for Ph.D Physics

FOR CMI EXAM PREPARATION

CMI 2016
Question Paper · Ph.D
Physics
EXAM YEAR TYPE SUBJECT

CMI 2016 Question Paper Ph.D Physics

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Instructions
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• Enter your Registration Number here: CMIPG ID

Enter your Examination Centre here:

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– The time allowed is 3 hours.
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– Total Marks: 100
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– Each question
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• Rough Work: The coloured blank pages are to be used for rough work only.

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CMI Ph. D. Physics Entrance Exam 2016

(1) A particle is constrained to move on a parabola y = αx2 in the plane where α is a constant
of dimension inverse length. Let there be a constant gravitational force in negative z direction.

(a) Write down the suitable generalized co-ordinates for this system. [1 Mark]
(b) Write down the Lagrangian and obtain Lagrangian equations of Motion. [1 Mark]
(c) Find the equilibrium position of the particle and write the equation for small ocscillations
about this equilibrium. [1 Mark]
(d) By solving the above equation show that particle executes simple harmonic motion about
the equilibrium position. [2 Marks]

(2) Consider three Newtonian particles of masses m1 , m2 , m3 which interact with each other
only through Newtonian gravitational force. Let their position vectors be ~r1 , ~r2 , and ~r3 respec-
tively with respect to the center of mass of the system.
(a) Write down the equations of motion for the system. [4 Marks]
(b) Assuming that the distance d between any pair of masses is equal and is in fact constant,
show that the system rotates with a constant angular velocity in it’s plane and about the center
of mass. [6 Marks]

(3) Consider a uniform hoop of mass M and radius R which hangs in a vertical plane supported
by a hinge at a point on the hoop as shown in the figure. Calculate the natural frequency of
small oscillations. [5 Marks]

(4) Recall that under a gauge transformation in magnetostatics, the vector potential is trans-
formed to A0 (r) = A(r) − ∇χ(r) where χ(r) is a scalar function. [20 Marks]

(a) Show that Ampere’s law (in differential form) takes the same form for both A and A0 .
[2 Marks]

(b) Give the differential equation that χ must satisfy so that A0 is in Coulomb gauge. Write
an integral expression for its solution, assuming A → 0 sufficiently fast as |r| → ∞. Proceed
by first writing Poisson’s equation for the electrostatic potential φ(r) due to a localized charge
distribution ρ(r) and its solution via an integral. Give an appropriate analogy between the two
problems. [5 Marks]

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(c) Find the magnetic field B(r) represented by the vector potential A = − 21 (r × b) where
b is a constant vector. Hint: For two vector fields we have ∇ × (A × B) = A(∇ · B) − B(∇ ·
A) + (B · ∇)A − (A · ∇)B. [5 Marks]

(d) Find whether the above vector potential A = − 12 (r × b) is in Coulomb gauge or not.
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Hint: ∇ · (A × B) = B · (∇ × A) − A · (∇ × B).
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[4 Marks]
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(e) Suppose the constant vector b = b0 ẑ points along the z axis. Find the vector potential
as
A. Roughly plot the resulting vector field A in the x − y plane.
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[4 Marks]
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(5)
(a) Consider a particle of mass m in a harmonic oscillator potential of frequency ω. Suppose
the particle is in the state
X∞
| α >= cn | n >
n=0

where
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cn = e−|α| /2 √
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and α is a complex number. Show that | α > is an eigenstate of the annihilation operator
â = mω
p i
(x̂ + mω p̂).
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(b) Show that the probability Pn (α) of finding the state | α > to have the value n when
the operator ↠â is measured is given by the Poisson distribution:
2n
−|α|2 | α |
Pn (α) = e
n!
[2]

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(c) Show that in the state | α >, < x >= Re(α) and < p >= 2m~ω Im(α). [2]
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[3]

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(e) If at a time t = 0, the state is | ψ(0) >=| α >, show that at a later time

| ψ(t) >= e 2 | αe−iωt >
−iωt

What is the origin of the phase factor? [3]

(f ) From (c) and (e) show that as functions of time, < x̂ > and < p̂ > follow the classical
trajectory of the harmonic oscillator. [2]

(g) Starting from â | α >= α | α >, show that the position space wave function for this
state is ψα (x) = ei<p̂>x/~ u0 (x− < x̂ >), where u0 (x) is the ground state gaussian function. [5]

(6) Consider a system of N free spin half particles of mass m obeying Fermi Dirac statistics at
temperature T=0 and occupying a volume V.

(a) Find the Fermi energy EF of the system. [3 Marks]

(b) How many particles have energy E ≤ EF ? [ 3 marks]

(c) Calculate the average value of U 3 where U is the speed of the particle. [4 marks]

(7). Consider a ideal gas having N distinguishable particles occupying a volume V. Consider a
small volume V1 .
(a) What is the probability that the volume V1 contains r particles? [3 marks]
(b) In the limit of large N show that r is maximum for r0 = N V1 /V . [3 marks]
(c) Show that the probability distribution of r − r0 is a gaussian [4 marks]
You can use the Stirling’s approximation ln N! = N ln N − N for the parts (b),(c) .

(8) Consider the integral Z 2π
dθ
I=
0 5 + 4 cos θ
iθ
(a) Define Z = e and express the integrand as a function of Z. [2 Marks]
(b) Identify the singularities of the above function in the complex Z plane. [2 Marks]
(c) Identify the contour of integration (in the complex Z plane) and perform the integral using
the residue theorem. [6 marks]

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(9) Given
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λ(t) = 1 − |t|, |t| ≤ 1
= 0, otherwise.

(a) Plot λ(t) [2 Marks]
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(b) Find the Fourier transform λ̃(ω) of λ(t). [6 Marks]
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(c) Plot λ̃(ω).
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[2 Marks]
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