UPSC Physics Optional PYQs (CSE & IFoS)PYQs

What is nuclear resonance absorption of gamma rays ? Describe the working and applications of Mossbauer spectrometer.

Discuss the rotational fine structure of electronic bands of molecule.

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

Describe in brief the Raman effect.

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

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

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}.

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}.

(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.

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

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

Explain the formation of molecular hydrogen in the interstellar medium.

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.

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 ?

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}.

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

Show that the square of the orbital angular momentum operator (L^2) commutes with any of the components of angular momentum operator L. Is it possible to measure L^2, L_x, L_y and L_z simultaneously ? Give reasons for your answer.

Find the energy of the particle of mass m moving in a potential field V(x) = \frac{2\hbar^2 b^2 x^2}{m} for which the time independent wave function is \psi(x) = \exp(-bx^2). Here b is a constant.

(ii) [L_z, L_+] = \hbar L_+ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z Prove that : (i) [L^2, L_z] = 0 (ii) [L_z, L_+] = \hbar L_+ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z where \hbar = \frac{h}{2\pi} (h is Planck's constant)

The ground state wave function of a harmonic oscillator is \psi_0(x) = \left(\frac{m\omega}{\hbar \pi}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right). (i) At which point is the probability density maximum ? (ii) What is the value of the maximum probability density ?