UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Why is the relaxation process so important in nuclear magnetic resonance (NMR)?

Why are {}^{12}\text{C} and {}^{16}\text{O} nuclei not suitable for the study of NMR?

Why does the electron paramagnetic resonance (EPR) spectroscopy utilize the microwave region of the electromagnetic spectrum, while nuclear magnetic resonance (NMR) uses radio waves?

Calculate the number of permitted electrons in a sub-shell and a shell in an atom.

The fundamental and 2^{\text{nd}} harmonic transitions of {}^{14}\text{N}^{16}\text{O} are recorded at 1876\cdot 06\text{ cm}^{-1} and 3724\cdot 2\text{ cm}^{-1}, respectively. Calculate the equilibrium vibrational frequency, the anharmonicity constant and the zero-point energy of the molecule.

{[Given : Mass of {}^{14}\text{N} = 23\cdot 25 \times 10^{-27}\text{ kg}, Mass of {}^{16}\text{O} = 26\cdot 56 \times 10^{-27}\text{ kg}]}

The OH-radical has a moment of inertia of 1\cdot 48 \times 10^{-46}\text{ kg}\cdot\text{m}^2. For J = 6, calculate its angular velocity and angular momentum. Find the energy absorbed in the J = 6 \rightarrow J = 7 transition.

Explain how Pauli's exclusion principle helps in determining the electronic configuration in a many-electron system.

What are hot bands in vibrational spectroscopy? Why are they called so?

Why can the experimental observation of the Lamb shift not be explained by the Dirac theory?

Find the positions of the first four rotational Raman lines in the spectrum of \text{H}_2, if its bond length is 0\cdot 742\text{ \AA}. What will be the effect of nuclear spin on the spectrum?

{[Given : Mass of {}^1\text{H} = 1\cdot 673 \times 10^{-27}\text{ kg}]}

In a Stern–Gerlach experiment, the magnetic field varies with distance in the z-direction according to \dfrac{dB_z}{dz} = 1\cdot 5\text{ T/mm}. The speed of silver atoms coming from the oven is 725\text{ m/s}. Silver atoms travel a distance of 3\text{ cm} through the magnet. Calculate the separation of the two beams of atoms when they leave the magnet.

{[Given : The mass of a silver atom = 1\cdot 8 \times 10^{-25}\text{ kg} and its magnetic moment is 1\text{ Bohr magneton}.]}

Explain why the rotational transition J = 0 \rightarrow J = 1 is often not the most intense.

Obtain the allowed eigenenergies of the half 1-D simple harmonic oscillator defined by the Hamiltonian H = \frac{p^2}{2m} + V(x), \quad \text{where} V(x) = \begin{cases} \dfrac{1}{2}m\omega^2 x^2 & \text{for } x \ge 0 \\ \infty & \text{for } x < 0 \end{cases} Express its eigenstates in terms of eigenstates of the full 1-D oscillator.

A particle is in the normalized state \psi which is a superposition of the energy eigenstates \psi_1 and \psi_2 with energies 10\text{ eV} and 30\text{ eV}, respectively. The average value of the energy of the particle in the state \psi is 24\text{ eV}. Find the state \psi in terms of \psi_1 and \psi_2.

Find the eigenvalues and eigenfunctions of a system described by Hamiltonian H = a e^{b \sigma_x}, where a, b are constants and \sigma_x is the x-component of Pauli matrix \vec{\sigma}.

If the position and momentum operators in 1-D are represented by \hat{x} and \hat{p} respectively, calculate the commutator [\hat{x}^2, \cos(\hat{p})].

A particle of mass m is in the ground state of a 1-D box of infinite potential of length a. If the length of the box is changed to 2a symmetrically without disturbing the wave function of the particle, find the probability that the particle will be in the ground state of the new box.

The de Broglie wavelength of a particle in thermal equilibrium at a temperature of 27\text{ }^\circ\text{C} is \lambda. At what temperature will it be \dfrac{\lambda}{2}?

The wave function of a spin \frac{1}{2} particle is given by \begin{pmatrix} -i \\ 2 \end{pmatrix}. What is the probability of finding the particle in the spin-up state?

The electron in the hydrogen atom is found in the state \psi(r, \theta, \phi) = A\, R(r) \sin\theta \cos\theta\, e^{-i\phi}. Find the z-component of angular momentum of the electron.