PHYSICS – 2003, PAPER II, SECTION A

                      Time Allowed: three Hours             Maximum Marks: 200

 Candidates should attempt Questions 1 and 5 which are compulsory, and any THREE of the remaining questions selecting at least ONE question from each Section.

Assume suitable date if considered necessary and indicate the same clearly.

LIST OFO USEFUL CONSTANTS

Mass of proton     = 1.673 x 10⁻²⁷ kg

Mass of neutron   = 1.675 x 10⁻²⁷ kg

Mass of electron    = 9.77 x 10⁻³¹ kg

Planck constant    = 6.626 x 10⁻³⁴ J s

Boltzmann constant   = 1.380 x 10⁻²³ J K⁻¹

Bohr magneton    = 9.273 x 10⁻²⁴ A/ m²

Electronic charge     = 1.602 x 10⁻¹⁹ C

Atomic mass unit (amu)      = 1.660 x 10⁻²⁷ kg

                                                     931 MeV

Velocity of light in vacuum, c     = 3 x 10⁸ m s⁻¹

m (⁴₂ He)         = 4.002603 amu

m (¹²₆ C)     = 12.00000 amu

 SECTION A

1.       Answer any four of the following :

     (a)    Define angular momentum, Express it in the operator form and show that

Lₓ Lᵧ - Lᵧ Lₓ = ih Lz

    (b)   A photon and an election have energy 15 keV each. Which of them will have a longer wavelength?

    (c)    Write down the Schrodinger equation in three dimensions for a free electron and solve it for electrons confined to a cube of edge L and hence obtain an expression for the density of states.

   (d)   State Franck- Condon principle and discuss its applications in molecular spectroscopy.

   (e)   If the magnetic moment of proton is 2.793 µN, calculate, giving necessary steps, the radio frequency at which nuclear magnetic resonance occurs in water kept in a uniform magnetic field of 2.4 T.

2.       (a) Consider a one-dimensional square –well potential, as shown below, which is attractive.

The potential is – Vₒ in the region x = a to x = -a and zero elsewhere. A stream of particles of mass M and energy E is directed from the left. Set up time-independent Schrordiner equation and obtain an expression for the transmission ratio from region I to II.

                        (b) Write the Schrodinger equation for a free particle confined in a cube of edge L. Determine the energy eigenvalues. Show that for a single value of energy different quantum states are possible. Explain the term degeneracy.

3.       (a) set up the time – independent Schrodinger equation for an election moving in a Coulob field V(r)  = - (ze² / 4ԑₒr), in polar coordinates. Solve the radial equation to get the energy eigenvalues.

                          (b) What is Zeeman Effect? How is can be understood on a quantum mechanical basis? Obtain an expression for the energy splitting.

4.       (a) Write the wave function for the ground state of the hydrogen molecule under the molecular orbital approximation. Estimate the energy of the election in the ground state of H⁺₂, molecule ion.

                          (b) Derive the combined vibration-rotation spectrum of a diatomic molecule. What are P and R branches?