LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – NOVEMBER 2003

PH 2801 / PH 821 – QUANTUM MECHANICS  I

29.10.2003                                                                                             Max.   : 100 Marks

1.00 – 4.00

SECTION – A

Answer ALL the questions.                                                                             (10 x 2 = 20)

1. Calculate the Compton shift in wavelength for an electromagnetic radiation of l = 6000 Å while the scattering angle is 30^o.

2. Find the de Broglie wavelength of an electron of energy 10 MeV.

3. Show that i (A⁺ – A) is a hermitian operator for any A.

4. Show that AB is hermitian only if [A,B] = 0 while A, B are hermitian.

5. Show that if any operator commutes with the parity operator, then the eigen functions of non-degenerate eigen values have definite parity.

6. If the probability densities are P₁, P₂, P₃ and P₄ for the do mains –a < x <- a/2; -a/2 < x < 0; 0 < x < a/2 ; a/2 < x <a respectively, what is P₁ + P₂ + P₃ + P₄ ?

7. Explain the basic assumptions of the perturbative technique.

8. Explain briefly WKB approximation.

9. Show that [Lx, Ly] = i  L₂.
10. Show that esatisfies the equation .

SECTION – B

Answer any FOUR questions.                                                                         (4 x 7.5 = 30)

11. Explain photo electric effect using the quantum theory of radiation.

12. State and prove Ehrenfest’s theorem.

13. (a) Explain the closure property.

(b) Give the physical interpretation of eigen values and eigen functions.

14. Obtain the energy eigen values for the single harmonic oscillator.

15. Explain the removal of degeneracy in a doubly degenerate case using time independent perturbation technique.

-2-

SECTION – C

Answer any FOUR questions.                                                                      (4 x 12.5 = 50)

16. (a) Explain the variation of heat capacity with temperature for solids. (6)

(b) Obtain an expression for the Compton shift.                                                    (6.5)

17. Explain the concept of quantum mechanical tunneling and show that the probability of barrier penetration is non-zero.

18. (a) State and prove Heisenberg’s uncertainly                                         (7.5)

(b) If [A, H] = 0, show that A becomes a constant of motion.                              (5)

19. (a) Express L² in spherical polar coordinates. (3)

(b) Solve the eigen value equation for L².                                                              (9.5)

20. Explain the ground state of Hydrogen molecule by using the variational technique.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – APRIL 2008

    PH 2806 / 2801 – QUANTUM MECHANICS – I

Date : 03/05/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

                                    PART A ( 10 X 2 = 20 MARKS )

ANSWER ALL QUESTIONS. EACH QUESTION CARRIES 2 MARKS.

1. What is meant  by classical approximation in wave mechanics ?
2. Can classical concepts explain the Compton effect ?
3. Define probability density and probability current density.
4. What are stationary states ?
5. What is an observable ? Give an example.
6. State the expansion postulate.
7. Sketch the first two wave functions of the stationary states of a simple harmonic oscillator.
8. What are coherent states ?
9. What is the effect of an electric field on the energy levels of an atom ?
10. What is the origin of the exchange interaction ?

PART B ( 4 X 7.5 = 30 MARKS )

ANSWER ANY FOUR QUESTIONS. EACH QUESTION CARRIES 7.5 MARKS.

11. State and explain the uncertainity principle.
12. (a) Explain Born’s interpretation of the wave function.

(b) Explain the significance of the equation of continuity.

13. (a) Explain the principle of superposition.

(b) Explain the property of closure.

14. Solve the eigenvalue equation for L ² by the method of separation of variables.
15. Explain the use of perturbation theory for the case of a 2-d harmonic oscillator.

PART C ( 4 X 12.5 = 50 MARKS )

ANSWER ANY FOUR QUESTIONS. EACH QUESTION CARRIES 12.5 MARKS.

16. Describe Compton effect and derive an expression for the shift in wavelength of the scattered beam.

17.Consider a square potential barrier on which is incident a beam of particles of energy E. Calculate the         reflected intensity and transmitted intensity, if the barrier height is V and width is a.

18. (a) Discuss the eigenvalue problem for the momentum operator.

(b) Discuss the postulate regarding evolution of a system with time.

19. Obtain the Schrodinger equation for a linear harmonic oscillator and find its eigenvalues and

eigenfunctions.

20. Discuss the WKB approximation method of solving eigenvalue problems. Consider the 1-d case and

find the solution near a turning point.

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