Reg. No.
APRIL 2022 U/1221/2037413/37623/24
QUANTUM MECHANICS
Time : Three hours Maximum : 100 marks
SECTION A — (10 × 2 = 20 marks)
Answer ALL questions.
1. What is the difference between classical and quantum
mechanics? Provide the equation relating the energy of emitted
radiation to frequency.
2. State Rayleigh-Jean law.
3. Distinguish Phase velocity and group velocity.
4. An electron in a molecule travels at a speed of 40m/s. The
uncertainty in the momentum p of the election is 10 6 of its
momentum. Compute the uncertainty in position x if the mass
of an electron is 9.1 × 10–31 kg using Heisenberg Uncertainty
Formula.
5. An eigenfiinction of the operator d 2 / dx 2 is e 2 x . Find the
corresponding eigenvalue.
6. State any two postulates of wave mechanics.
7. Show L2 x L2 y (i Lx iLy ) (iLx iLy) .
8. What is the expectation value of L2?
9. Calculate the energy difference between the ground state and
the first excited state for an electron in one dimensional rigid
box of length 10 10 m.
10. Mention the significance of zero point energy.
SECTION B — (5 × 7 = 35 marks)
Answer ALL questions.
11. (a) Illustrate how Classical Physics failed to account for the
spectral distribution of energy density in the black-body
radiation. How did Planck overcome the difficulty?
Or
(b) Describe Lummer and Pringsheim experiment for the
study of black body radiation. Explain the results of the
experiment.
12. (a) Derive an equation for de Broglie wavelength of matter
particle in terms of kinetic energy and temperature. Show
that the circumference of the Bohr orbit for the hydrogen
atom is an integral multiple of the de Broglie wavelength
associated with the electron revolving around the orbit.
Or
(b) Describe Davisson and Germer experiment for the study of
electron diffraction. Interpret the results in detail.
13. (a) Derive Schrodinger’s equation. What is the significance of
the wave function? Calculate the expectation value px of
the momentum of a particle trapped in a one – dimensional
box. (2+2+3)
Or
(b) Define linear operators. Find the value of the constant B
that makes e ax 2 an eigen function of the operator
d 2 / dx 2 B x 2 . What is the corresponding eigen value?
(2+5)
14. (a) Deduce the commutation relation for the components
Lx , L y , Lz of the orbital angular momentum and show that
2 2 2
all the three components commute with L2 LX LY Lz ,
derive eigen values of L2 and Lz.
Or
(b) From the classical angular momentum operator L r p
and the commutation rules for r and p operator derive the
commutation rules for the operators Lx, Ly, Lz.
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15. (a) Calculate the values of the energy of a particle in a one
dimensional box. Indicate graphically the first three wave
functions for such a particle.
Or
(b) Write the Schrodinger’s wave equation of the hydrogen
atom in polar coordinates. Explain the origin and
significance of the quantum state n, 1 and m1.
SECTION C — (3 × 15 = 45 marks)
Answer ALL questions. (Q. No. 16 is compulsory).
16. Describe in detail the experiment of G.P. Thomson on the
diffraction of electrons. Elucidate the results obtained.
17. (a) Derive Planck’s radiation law and its distribution function.
Using Planck’s radiation law, prove that the total energy
density Etotal is given by Etot aT 4 where
a = 8 5 k 4 / 15 h3 c3 .
Or
(b) State the postulate of Bohr atom model. Obtain the
expressions for the radius and electron energy of the nth
orbit. Explain how Bohr’s atom model successfully accounts
for the hydrogen spectrum.
18. (a) Give an account of Heisenberg’s uncertainty Principle.
Outline an idealised experiments to bring out its
significance. The lifetime of an excited state of an atom is
about 10–8 S. Calculate the minimum uncertainty in the
determination of the energy of the excited state.
Or
(b) Exemplify the principles on which electron microscope
works. Compare this instrument with an optical
microscope.
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