For transition to the ground state what is the longest wavelength that can be emitted by hydrogen ?

(i) State and explain Franck-Condon principle. (5) (ii) In three separate pairs of electronic states (one ground and another upper state) of diatomic molecules, the average internuclear distances are -

1. equal;
2. slightly greater;
3. appreciably greater. In accordance with Franck-Condon principle, obtain most favorable transitions from the ground state vibrational levels in each of the above situations. Illustrate the transitions by drawing suitable energy-level diagrams. (15)

Describe Stern-Gerlach experiment. Explain how it demonstrates the discrete nature of the magnetic moment of an atom.

(i) What is Lamb shift ? Has it been observed in any atom other than hydrogen? (10) (ii) How does the 21 cm hydrogen line originate? Comment on its astronomical applications. (10)

Write an expression for the energy of a diatomic molecule executing both rotation and vibration in a given electronic state. Explain the observed spectra giving a schematic diagram. What are O, P, Q, R and S branches ?

What are the gyromagnetic factors of the three cases, viz., a spinning electron, an orbiting electron and a free proton ?

Determine the order of magnitude of the energy involved in the spin-orbit interaction of 2p state of an atom with an electron. Given the magnetic field due to the state is 0.28 T.

Explain and discuss the basic principle of nuclear magnetic resonance (NMR).

State and discuss the Franck-Condon principle giving special reference to its applications.

Examine whether the molecule ^{11}\text{B}^{16}\text{O} can show -- (i) a pure rotation spectrum ; (ii) a vibration-rotation spectrum. Give reasons.

(i) How many lines occur in the multiplet arising from ^{2}\text{P}_{3/2} \rightarrow {^{2}\text{S}_{1/2}} and ^{2}\text{P}_{1/2} \rightarrow {^{2}\text{S}_{1/2}} transitions of alkali metal atoms placed in a weak magnetic field, why ? (ii) The wavefunction of a particle confined in a cube of volume L^3 is given by \Psi (x,y,z) = \left( \frac{2}{L} \right)^{3/2} \sin \frac{\pi x}{L} \sin \frac{\pi y}{L} \sin \frac{\pi z}{L} Calculate the average values of p_x and p_x^2 in the region 0 < x < L.

Obtain an expression for the vibration-rotation energy levels of a diatomic molecule in a given electronic state. The wave numbers of the vibrational transitions occuring in HF, HCl and HI molecules are 4141 \cdot 3 \text{ cm}^{-1}, 2988 \cdot 9 \text{ cm}^{-1} and 2309 \cdot 5 \text{ cm}^{-1} respectively. Compare the force constants of these three molecules.

(i) Explain the molecular phenomenon of spontaneous emission between two electronic states of the same multiplicity and differentiate it form that of different multiplicity. (ii) The constants B and V_0 for a KCl molecule have values 1 \cdot 43 \times 10^{-5} \text{ eV} 8 \cdot 4 \times 10^{12} \text{ s}^{-1} respectively. Determine the number of rotational levels between the vibrational levels v = 0 and v = 1.

Describe Stern-Gerlach experiment and discuss its implications.

Obtain an expression for the vibration-rotation energy levels of a diatomic molecule in a given electronic state. The wave numbers of the vibrational transitions occuring in HF, HCl and HI molecules are 4141.3\text{ cm}^{-1}, 2988.9\text{ cm}^{-1} and 2309.5\text{ cm}^{-1} respectively. Compare the force constants of these three molecules.

Describe Stern-Gerlach experiment and discuss its implications.

(i) Calculate the wavelength of neutrons at a temperature of 20^\circ\text{C}. (ii) What are the term symbols for atoms with the following S (total spin) and L (total orbital) quantum numbers, S = 1/2, L = 3 ?

(i) Calculate the Zeeman shift observed in the normal Zeeman effect when a spectral line of wavelength 5000\text{ \AA} is subjected to the magnetic field of 0.4\text{ Wb m}^{-2}. (ii) Show that for the ground state of hydrogen atom the mean value of r is 3/2\text{ a}_0, where \text{a}_0 being Bohr radius constant.

Define the gyromagnetic ratio and obtain an expression for the Lande g factor. What is the importance of g factor in spectroscopy?

Discuss the salient features of O, P, Q, R and S branches of electronic spectrum of diatomic molecules. Which spectrum contains only branches, both branches and bands, branches and progressions ?