A radionuclide (N_1) decays to a radioactive daughter (N_2) that subsequently decays to a third daughter (N_3). If at t = 0, the initial concentrations of N_1, N_2 and N_3 are N_1^0, 0 and 0 respectively, and the decay constants are \lambda_1 (from N_1 \rightarrow N_2) and \lambda_2 (from N_2 \rightarrow N_3), then derive the expression for N_2 at time t. Given that the daughter N_3 is stable.

What is Lamb shift in the fine structure of hydrogen spectrum ? Discuss its theory based upon second quantization.

Explain how the hydrogen spectrum is used for imaging the universe.

Differentiate between L-S coupling and J-J coupling. What are the possible orientations of \vec{J} for the J = \frac{3}{2} and J = \frac{1}{2} states that correspond to l = 1 ?

Discuss the rotational fine structure of electronic bands of molecule.

Describe in brief the Raman effect.

(i) Using free electron theory of metals, calculate the Fermi energy level of sodium atom at absolute zero. Assume that sodium has one free electron per atom and its density is 0\cdot 97\text{ gm/cm}^3. (ii) Draw the energy level diagram and mathematical expressions for the following : I. E_n of an electron confined in a one-dimensional box II. Linear harmonic oscillator Make a qualitative comparison of the above two cases.

A beam of hydrogen atoms in a Stern-Gerlach experiment obtained from an oven heated to a temperature of 400\text{ K} passes through a magnetic field of length 1\text{ m} and having a gradient of 10\text{ T/m} perpendicular to the beam. Calculate the transverse deflection of an atom of the beam at a point where the beam leaves the field. The value of Bohr magneton \mu_B is 9\cdot 27 \times 10^{-24}\text{ A m}^2 and the Boltzmann constant k is 1\cdot 38 \times 10^{-23}\text{ J/K}.

Explain the formation of molecular hydrogen in the interstellar medium.

How is Rydberg constant related to emission wavelength of hydrogen spectrum ?

In the vibrational Raman spectrum of HF, the Raman lines are observed at wavelengths 2670\text{ \AA} and 3430\text{ \AA}. Find the fundamental vibrational frequency of the molecule.

Describe Electron Paramagnetic Resonance. Highlight its differences with NMR and discuss its applications.

Calculate the possible angles between \vec{L} and \vec{S} for a d-electron in one-electron atom.

Show that for a diatomic molecule with two nuclei of mass 'M' separated by a distance 'a', the rotational energy of nuclear motion is lower than electronic energy by a factor of \frac{m_e}{M}.

How do Stokes lines appear in Raman spectrum as per classical and quantum theory of Raman effect ?

The force constant of the bond in CO molecule is 1900\text{ N m}^{-1}. Calculate the energy of the lowest vibrational level. The reduced mass of CO molecule is 1\cdot 14 \times 10^{-26}\text{ kg}. Given h = 6\cdot 63 \times 10^{-34}\text{ J s} and 1\text{ eV} = 1\cdot 6 \times 10^{-19}\text{ J}.

$, [J_z^2, J_y] and [J^2, J_y], and then show that \langle J, m | J_x^2 | J, m \rangle = \langle J, m | J_y^2 | J, m \rangle

Find the energy levels of a spin S = \frac{3}{2} particle whose Hamiltonian is given by H = \frac{\alpha}{\hbar^2}(S_x^2 + S_y^2 - 2S_z^2) - \frac{\beta}{\hbar} S_z where \alpha, \beta are constants. Are these levels degenerate?

A particle limited to the x-axis has the wave function \phi(x) = bx^2 between x = 0 and x = 2; the wave function \phi(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1.0 and x = 1.5. (ii) Find the expectation value <x> of the particle position.

Consider a particle of mass m moving freely between x = 0 and x = a inside an infinite square well potential. Calculate the expectation values \langle x \rangle_n, \langle p \rangle_n, \langle x^2 \rangle_n and \langle p^2 \rangle_n, and compare them with their classical counterparts.