JLR-07/10
PHYSICS
Time allowed: Three Hours
Full Marks: 100
All questions carry equal marks.
Answer any FIVE questions.
I. Using the Residue theorem, evaluate the integral,
00 p_1
I-Ix
6
.
o
Find all square roots ofthe matrix,
3/2 3/2 . 6
Solve the following equation using Laplace transform

t F t 0
where I and =O. (The prime symbol .indicates differentiation w.r.t. t). 8
2. Solve the problem of linear harmonic oscillator using Hamilton Jacobi equation. 5

Find the total time derivative of H in terms ofPoisson bracket, where the symbols have their usual meaning. 5


Give the relativistic generalization ofNewton's second


law and deduce the relation E mc2• 6+4 MSH-5329 1 (Coned.)
3. State the postulate(s) of non-relativistic quantum
mechanics conceming the results ofmeasurement of
an observable. 5
Find the position operator in the momentum
representation and verify that it is Hermitian. 5
State and prove Wigner-Eckert theorem. 2+8
4. What is negative temperature? Find an expression
for the specific heat at constant volume ofa system
which exhibit negative temperature and discuss its
temperature dependence.
Starting from the grand canonical partition function,
obtain an expression for the energy density in the
frequency interval v to v dv ofblack body radiation.
Hence find the expressions offree energy and pressure
of black body radiation. 6+4
5. Derive an expression for the specific heat of a free
electron gas and show that at low temperature the
specific heat is proportional to temperature. 8
Derive the energy dispersion relation for lattice
vibrations of a I-dimensional chain of atoms. 6
Give an account of the Weiss theory. 6
6. Discuss the vector model of LS coupling for non­
equivalent valance electrons. 7
What do you mean by ''Population inversion" Why
is it necessary for laser radiation? 6
Explain the physics of holography. 7

MSH-5329 2 (Contd.)
7. Asswning the nucleus to be a degenerate Fermi gas
ofZ protons and N neutrons, find an expression for
the total zero point kinetic energy ofthe nucleus.
10
Following Fermi's theory ofbeta decay derive an
expression for the time rate of emission probability
of an electron with momentum within a range
p to p+dp. 10
8. Consider a fixed bias circuit ofan transistor
(silicon) in CE configuration where the collector and
the base are connected to 10 volt dc supply through
two resistors of magnitude 5 kQ and 20 kQ
respectively; the emitter is connected to the common
point. The d.c. current gain ofthe transistor is 80.
Find the values ofIe and Is it possible to use the
circuit as voltage amplifier? Explain your answer.
10
With a neat circuit diagram, explain the operation of
a class-B push-full amplifier. 10
9. Deduce the Newton-Raphson formula to find a root
ofthe equation numerically. 10
Write a C program to integrate a given function
from a given value c to another given value
d using Simpson's rule. 10

MSlJ,-5329 3 (Contd.)
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