}

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

}

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

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

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?

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.

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

}

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

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

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?

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}.

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.

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.

$.

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 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}?

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.