UPSC Physics Optional PYQs (CSE & IFoS)PYQs

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

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}]}

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

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

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

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

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

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}]}

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

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 is in the ground state of a 1-D simple harmonic oscillator potential, V(x) = \dfrac{1}{2}m\omega^2 x^2. Calculate the position uncertainty of the particle.

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.

The wave function of a hydrogen atom is written as \psi_{n, l, m}(r, \theta, \phi). Explain the quantum numbers n, l, m and mention their ranges. Show that \psi_{n, l, m}(r, \theta, \phi) is n^2 degenerate.

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.

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}?