Kerala University 2006-2nd Sem M.Sc Physics Iester , - Question Paper

Reg. No..................................................................(Pages: 2)    8261 Name:.............................

III Semester M.Sc. Degree Examination, November 2006 Branch - II: PHYSICS PH 231 : Quantum Mechanics

Time: 3 Hours    Max. Marks: 75

PART - A

Answer any five questions. Each question carries 3 marks.

1. a) Show that the components of orbital angular momentum operators satisfy the relation LxL = iL.

b) What is meant by adiabatic approximation ?

_C) state Wigner - Eckast theorem and explain its significance.

d)    State and explain Femois Golden rule.

e)    Write a note on Lamb Shift.    *

f)    Show that YjxYuYji = ~²Yy where Yn_? Yu are Diracs gamma matrices. (There is summation over repeated indices.)

g)    Show that [a_k, N_k] = a_k and [a+, nJ = -a+ where a_k,a + andN_k are the bosonic annhilation, creation and number operators.    (5x3=15 Marks)

PART - B

Answer all questions. Each carries 15 marks.

2. a) i) Obtain the common Eigen states of the angular momentum operators j2 & j_z f_{or a} particle. Comment on the nature of the eigen values.

ii) Show that J₊ J_ = J² +h J_z - J? where J=J_X J_y.

OR

P.T.O.

b) Use the first order perturbation theory to find out the energy levels of the ground state of the Helium atom. How are the results modified if one uses the variation

technique ?

3.    a) What is Born approximation ? Apply it to obtain the differential cross section

for a square well potential and discuss its validity.

,    OR

b) i) What is meant by electric dipole transition ?

ii) Obtain expressions for transition probability of spontaneous and induced emission of radiation for such transitions.

4.    a) i) Deduce the Dirac equation for a free particle. Show how the relations of

this equation predict the existence of positron, ii) Show that the Dirac a and(3 matrices need to be at least 4x4 matrices.

OR

b) i) Explain the principle of indistinguishability of identical particles. Considering the case of a system of two identical particles show that the wave function is

either symmetric or antisymmetric, ii) Taking the atom as an example show that the singlet state is always higher in ~ energy than the "triplet stateT    - - -- (3x15=45 Marks)

PART - C

Answer any ithrce questions. Each question carries 5 marks.

5. a) Evaluate the Clebsch-Gordon Coefficient C(121; 1-2-1) using its symmetry properties given C(112: 112) = 1.

b)    Assuming that a perturbation H¹ = CX (C being a constant) is applied to a particle in a one dimensional box of side L. Show that the first order correction

to its energy is .

c)    Show that the zero energy scattering cross section for scattering by hard sphere

of radiusais 4rca².

d)    Write down the different spin wave functions for a two electro system whose

interaction is negligible.

e)    Obtain the canonically conjugate momentum density and the Hamiltonian density

for the given Lagrangian field density L = A(j)² + B( (j))²; \x = 1,2,3,4 forthe

scalar field <t>. Treat A and B as constants.    (3x5=15 Marks)

Attachment: kerala-university-2006-m-sc-physics-40181-1.pdf

Earning:   Approval pending.