END TERM EXAM SPRING-2006-07

PH-102 (B. Tech. I)

Time: 3 hours M. M.: 70

Weightage: 50%

Note: (i) Attempt all parts of question 1.

(ii) Attempt one out of the two parts ((a) and (b)) of each of the remaining four questions.

  1. (a) Using the uncertainty principle, show that the Bohr’s radius of Hydrogen atom in its first

orbit can be estimated as r=ε0 h2/(π m e2).[3]

(b) A substance has fcc lattice structure, and a density of 6250 kg m-3 and atomic weight 60.2

amu. Find the free volume in a unit cell of this substance. [3]

(c) A particle in a one-dimensional box, of width 25 Å, is in its ground state. Find its average

position and momentum. What is the probability of finding it within an interval of ± 0.2 Å

about the center of the box?[4]

(d) Explain clearly and briefly how Davisson and Germer analysed and demonstrated the wave-

particle duality through their experiment. [4]

(e) Describe briefly what type I and II superconductors are and what the Meissner effect is.

A superconducting material has a critical temperature of 4 K and a critical field of 0.036 T

at 0K. Find the critical field at 2K.[4]

(f) Discuss analytically the low and high temperature behavior of the specific heat of solids

through Einstein’s theory (without deriving the required expression). Also discuss the

limitations of this theory.[4]

  1. (a) Write the Schrödinger equation for a one-dimensional harmonic oscillator and the resulting

wave functions for the ground and the first two excited states and their energy eigenvalues.

Plot these wave functions and eigenvalues. Can the quatum mechanical results be compared

with the classical results?[6+4+2]

OR

(b) Describe the Stern-Gerlach experiment and explain how it shows space quantization of the

spin angular momentum. Discuss the fine structre of the Hα-line.[6+6]

  1. (a) Differentiate between normal and anomalous Zeeman effects with suitable examples using

energy level diagrams. Show that the difference between successive split energy levels

corresponding to 2P3/2 is twice the separation between successive split energy levels

corresponding to 2P1/2in any weak magnetic field. [6+6]

OR

(b) Calculate the ratio of Einstein’s A and B coefficients and interpret your result. Explain the

principle and working of a He-Ne laser using proper energy level diagram. Discuss briefly

the principle of holography.[3+6+3]

  1. (a) Using the solutions of the Schrödinger equation for free electrons in a three-dimensional

conducting solid, obtain expressions for Fermi energy and density of states. Calculate the

average energy of the electrons at 0K. [8+4]

OR

(b) Derive expression for paramagnetic susceptibility and also obtain the expression for the

susceptibility of a ferromagnetic material.Show the variation of both susceptibilities with

temperature.[6+3+3]

5. (a) Obtain the expression for electrical conductivity of metals. Show its relation with thermal

conductivity. The Fermi energy of Potassium is 2.1 eV and its electrical conductivity is

1.5 x 107 Ω-1 cm-1. Calculate the mean free path of electrons in Potassium. [6+3+3]

OR

(b)Obtain the semi-empirical binding energy formulafor nuclei.Explain the significance

and origin of each term appearing in the formula. Suppose there is a very heavy nucleus

with mass number A and atomic number Z which fissions into two identical nuclei each

having mass number A/2 and atomic number Z/2 . Using the above binding energy formula

and disregarding the pairing term, show that the energy released in this process is

approximately given by the expression:

E ≈ asurface A2/3 (1 – 21/3) c2 + aCoulomb Z2 A-1/3 (1 – 2-2/3). [8+4]

*****

Charge of electron = 1.6 x 10-19 C

Planck’s constant = 6.6 x 10-34 Js

mass of the electron= 0.00055amu

Bohr magneton= 9.27 x 10-24 JT-1

Boltzmann constant= 1.38 x 10-23 JK-1

Avogadro’s number = 6.023 x1026 (kmol)-1