Download LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – PHYSICS
SUPPLEMENTARY EXAMINATION – JUNE 2007
PH 2808 - QUANTUM MECHANICS
Date & Time: 25/06/2007 / 9:00 - 12:00
Dept. No.
PART- A
Max. : 100 Marks
(10 x 2m = 20m)
1. State and explain any two admissibility conditions on the quantum mechanical
wave function.
2. Prove explicitly that the momentum operator is a self-adjoint operator.
3. Write down the ground state energy eigenfunction of a simple harmonic oscillator?
Sketch its graph.
4. Define the parity operator by its effect on a wave function. What are its
eigenvalues?
5. If A is any Hermitian operator and  is a real number, prove that exp ( iA ) is
unitary.
6. What is a projection operator?
7. Write down the expression for Hamiltonian of spin-orbit interaction of an electron
subject to an spherically symmetric electric potential.
8. What are Clebsch-Gordon coefficient?
9. “The second order correction to the energy eigenvalue (in perturbation theory) is
necessarily negative if the unperturbed state is the ground state “. Justify this
statement.
10. What is the basis of the WKB approximation and why is it referred to as the semiclassical approximation method?
PART- B
(4x7 1/2m= 30 m)
ANSWER ANY FOUR QUESTIONS
11. A) Prove that the eigenvalues of a self-adjoint operator are real and any two
eigenfunctions belonging to distinct eigenvalues of a self-adjoint operator are
mutually orthogonal
B) Explain the Schmidt orthogonalisation procedure
(4 ½ + 3)
12.Discuss the behaviour of the radial wave function of a particle in a central potential,
both near the origin and in the asymptotic region
13.Obtain the Schoredinger representations for the position and the momentum
operators based on the general representation theory.
14. Prove the following properties of the Pauli matrices: a) 2 = 3,
b) xy = - yx and c) +2 = 0.
15. Discuss the WKB approximation method for the one-dimensional Schoredinger
equation and obtain the asymptotic nature of the solution.
PART- C
(4x 12 1/2m= 50 m)
ANSWER ANY FOUR QUESTIONS
16. A) Derive the uncertainty relation for a pair of non-commuting observables.
B) Given that  x,p  = ih, obtain the value of  x2, p2 .
(8+41/2)
17. Obtain the energy eigenvalues and energy eigenfunctions of a simple harmonic
oscillator by the ladder operator method.
18. A) Prove that the momentum operator in quantum mechanics is proportional to the
generator of infinitesimal translations.
B) Explain the Schoredinger picture of time evolution of a quantum mechanical
System
(8+41/2)
19. Starting from the commutation relations between the components of the angular
momentum operator J, obtain the matrix representations for J 2 and J+ in the  jm
basis for j = 1.
20. Discuss the time independent perturbation theory to obtain the first order
corrections to both the energy eigenvalue and the energy eigenfunction of a state of
a non-degenerate quantum mechanical system.
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