What is Franck-Condon principle? Discuss the intensity distribution in the vibrational electronic spectra of a diatomic molecule on the basis of this principle.

If the state of the hydrogen atom is 2p state, calculate the energy levels of the spin-orbit interaction Hamiltonian A \vec{L} \cdot \vec{S}, where A is a constant.

What is Lamb shift? Why does the study of Lamb shift enjoy so much attention?

Discuss the anomalous magnetic splitting of sodium D-lines with necessary illustration (no deduction is required). Identify the polarization of each component line. What condition needs to be satisfied for this splitting to take place?

What is Lamb shift? Discuss its significance in determining the fine structure of \mathrm{H}_{\alpha} Balmer line in hydrogen atom.

Draw the schematic diagram of a nuclear magnetic resonance (NMR) spectrometer. Discuss nuclear magnetic resonance imaging.

Write down the characteristics of Raman spectral lines. Explain why Stokes lines are more intense than anti-Stokes lines. With exciting line at 2536~\text{\AA}, a Raman line for a sample is observed at 2612~\text{\AA}. Calculate the Raman shift in wave number.

The first rotational line of ^{12}\text{C}^{16}\text{O} is observed at 3\cdot 84235~\text{cm}^{-1} and that of ^{13}\text{C}^{16}\text{O} at 3\cdot 67337~\text{cm}^{-1}. Assuming the mass of ^{16}\text{O} to be 15\cdot 9949~\text{u}, calculate that of ^{13}\text{C}.

Write down the Franck-Condon principle and explain how it accounts for the intensities of spectral lines in vibrational-electronic spectra with suitable illustrations.

How do you define density of states? Show that the density of states with wave vector less than \vec{k} in a three-dimensional cubic box of volume V can be given by D(\omega)=\frac{V}{2\pi^2}k^2\left(\frac{dk}{d\omega}\right) in the frequency spectrum between \omega and \omega+d\omega. Here, assume that the number of modes per unit range of k is L/(2\pi), L being the length of each side of the cubic box.

Write down the Hamiltonian operator for a linear harmonic oscillator. Show that the energy eigenvalue of the same can be given by E_n=\left(n+\frac{1}{2}\right)\hbar\omega_0 at energy state n with \omega_0 being the natural frequency of vibration of the linear oscillator. Prove that n=0 energy state has a wave function of typical Gaussian form.

An electron is confined in the ground state of a one-dimensional harmonic oscillator such that \Delta x = 10^{-10}~\text{m}. Assuming \langle T \rangle = \langle V \rangle, find the energy in eV required to excite it to the first excited state.

Define Pauli spin matrices \sigma_x, \sigma_y and \sigma_z. Using these definitions, prove the following:

(i) \sigma_x^2=\sigma_y^2=\sigma_z^2=1

(ii) \sigma_x\sigma_y=i\sigma_z;\ \sigma_z\sigma_x=i\sigma_y;\ \sigma_y\sigma_z=i\sigma_x

$. Hence find the uncertainty product (\Delta x)(\Delta H).

Define angular momentum of a particle and find out the three components of the angular momentum operator \hat{L} in Cartesian coordinates. Show that \hat{L}^2=-\hbar^2\left[r^2\nabla^2-\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)\right] Prove that the operator \hat{L}^2 can also be expressed as \hat{L}^2=-\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] in spherical polar coordinates (r,\theta,\phi).

= 2i\sigma_z$ and (ii) \sigma_x \sigma_y \sigma_z = i.

Show that the mass and linear momentum of a quantum mechanical particle can be given by m=h/(\lambda v) and p=h/\lambda, respectively, where h, \lambda and v are Planck’s constant, wavelength, and velocity of the particle, respectively. Comment on the wave-particle duality from these relations.

Define mathematically the Bohr radius of a hydrogen atom and show that the binding energy at state n of this atom can be given by E_n=-\frac{1}{2}\frac{Ze^2}{(a/Z)4n^2\pi\epsilon_0} where Z is the atomic number of H atom. Calculate the numerical values of a and E_1 of H atom.

Describe normal and anomalous Zeeman effect. Explain how it lifts the degeneracy in hydrogen atom.

In the |jm\rangle basis formed by the eigenkets of J^2 and J_z, show that \langle jm| J_- J_+ |jm\rangle = (j-m)(j+m+1)\hbar^2 where J_+ = J_x + i J_y and J_- = J_x - i J_y.