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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