JNU-Ph.D Entrance 2014
Uploaded by
ken adamsJNU-Ph.D Entrance 2014
Uploaded by
ken adamsfiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Pre Ph.D. / Ph.D. Entrance 2014
PART – A
Note: Answer all questions. Each question carries 6 marks.
2 d
Q1. Using the method of residues, calculate the integral
0 5 3 cos
L
Q2. A particle of mass m moves in a one-dimensional box located between x and
2
. Thus the potential function V x 0 if x but is infinite otherwise. Suppose,
L L
x
2 2
L2
at t 0 , the wave function of the particle inside the box is given by x x 2 ,
4
where is a suitable normalization factor.
(a) Calculate the value of .
(b) Calculate the wave function of the particle for t 0 .
Q3. A block B of mass m is lying on the frictionless top surface of a triangular block C of
mass M as shown in the figure below. The flat lower surface of C is resting on a
frictionless horizontal surface.
(a) Write down the Lagrangian of this system of two masses.
(b) Derive the Lagrange’s equation(s) of motion.
(c) Suppose both B and C are held at rest initially. Then, at t 0 , we release them so
that B can slide freely down the surface of C under the action of gravity and C is also
free to recoil. Solve the Lagrange’s equation(s) of motion to find how the vertical
coordinate of the block B will change with time.
B
Q4. Consider a spherical shell with inner radius a and outer radius b . Throughout this shell,
there is electrostatic charge with uniform charge density .
(a) Calculate the electric field for every value of r , the radial coordinate.
(b) Calculate the electrostatic potential for r a .
Head office Branch office
fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics,
Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT
Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 1
fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Q5. Consider a one-dimensional quantum harmonic oscillator of frequency v in equilibrium
with a heat bath at temperature T . Calculate the partition function, average energy and
root mean square fluctuation of energy for this system.
PART – B
Note: Answer all questions. Each question carries 4 marks.
Q1. For dimensional analysis, we usually take mass, length and time as the basic physical
variables with dimensions M , L and T . Suppose, deviating from this normal practice,
we choose to use speed, angular momentum and frequency as the basic variables and
denote their dimensions by E , P and H . What would be the dimensions of linear
momentum and torque in terms of E , P and H ?
Q2. A radioactive element A decays to B with the decay rate A . In turn, B decays to C
with the decay rate B . Suppose, initially, only A -type nuclei are present in such a
radioactive sample. After what time would the number of B -type nuclei in this sample
reach its maximum value?
1
Q3. For a free relativistic particle of rest mass m the Hamiltonian is H p. pc 2 m2c 4 2
.
Here p is the momentum vector and c is the speed of light in vacuum. Derive the
expression of the Lagrangian for this particle.
x
dxe x dye y .
2 2
Q4. Evaluate the integral
0 0
1
Q5. A spin- particle is in the spin state
2
1 1 1 1
cos ei , sin ,
2 2 2 2 2 2
1 1 1 1
, and , are the eigenstates of S z , the operator for the z -component of spin
2 2 2 2
angular momentum, with eigenvalues of and , respectively. Calculate the
2 2
expectation values of S x and S y in the state .
Head office Branch office
fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics,
Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT
Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 2
fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Q6. A metallic block of mass 2 kg and specific heat 2.5 cal / K per gram is initially at a
temperature of 80 o C . It is then dropped into a swimming pool, the temperature of this
pool being 25 o C . How much will the entropy of this combined system of the metallic
block and the swimming pool change by the time thermal equilibrium is established?
Does the entropy increase or decrease?
Q7. Consider a divalent metallic element in a crystalline solid state with a simple cubic
primitive cell of side 4 angstrom. In the free electron approximation, what is the length
of the Fermi wave vector for this metal?
Q8. In the following inverting feedback circuit of an operation amplifier, calculate the voltage
gain. Take R1 R2 R4 100k , R3 RL 10k .
R2 R4
R3
R1
vo
vi RL
Q9. Calculate the binding energy of a positronium, which is a bound state formed by an
electron and its anti-particle (positron). [You may use any results derived in standard
introductory textbooks on Quantum Mechanics]
Q10. Consider an electron moving with a kinetic energy of 20 MeV . By what percentage is its
speed different from c (the speed of light in vacuum)?
Head office Branch office
fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics,
Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT
Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 3