PYQ Bank

List in two separate columns, the quantities that are conserved and not conserved in the weak interaction of particles.

(i) Plot proton number (Z) versus neutron number (N). What is the inference from the plot with regards to light and heavy nuclides ? (ii) Which nucleus would you expect to be more stable, ^{7}_{3}\text{Li} or ^{8}_{3}\text{Li} ? Justify your answer.

Calculate in terms of the nuclear magneton, \mu_{\mathrm{N}}, the magnetic dipole moment of ^{3}S_{1} state of deuteron. Given, \mu_{\mathrm{p}}=2\cdot792847\mu_{\mathrm{N}} and \mu_{\mathrm{n}}=-1.913042\mu_{\mathrm{N}}.

Explain how the magnetic moments of atoms, the space quantization of angular momentum and the spin of electron are measured using Stern--Gerlach experiment.

Explain the mechanism of Fluorescent emission of radiation. Distinguish between Fluorescence spectra and Raman spectra of a diatomic molecule.

A Raman line associated with a vibrational mode which is both Raman and infrared active is found at 4600\text{ \AA} when excited by light of wavelength 4358\text{ \AA}. Calculate the wavelength of the corresponding infrared band.

Obtain an expression for the rotational energy levels of a diatomic molecule, taking it as a rigid rotator. Discuss its spectrum and the selection rule.

Write the principle of nuclear magnetic resonance (NMR). Explain the design and working of NMR, and write its important applications.

Explain the principle and working of Stern-Gerlach Experiment.

Determine the normal Zeeman splitting of the cadmium red line of 6438\,\mathring{\mathrm{A}}, when the atoms are places in a magnetic field of 0.009\,\mathrm{T}.

The calcium line of wavelength \lambda = 4226\cdot 73\text{ \AA} (\text{P} \rightarrow \text{S}) exhibits normal Zeeman splitting when placed in a uniform magnetic field of 4\text{ Weber/metre}^3. Calculate the wavelength of three components of normal Zeeman pattern and the separation between them.

Calculate the frequency of the first Bohr orbit of hydrogen atom.

The potential energy of a diatomic molecule in terms of the interatomic spacing R is given by U(R)=-\frac{A}{R^2}+\frac{B}{R^{10}} where A=1.44\times10^{-3}\,\mathrm{J\,m^2} and B=2.19\times10^{-115}\,\mathrm{J\,m^{10}}. Calculate the equilibrium spacing, R_e and the dissociation energy.

(i) State and explain position momentum uncertainty principle. Justify that this principle is not just a negative statement rather a useful tool, with one example. (ii) The lifetime of an excited state of an atom is about 10^{-8}\text{ sec.} Calculate the minimum uncertainty in the energy of the excited state.

(i) What do you mean by expectation value of a physical quantity ? How does it help to extract information from a wave function ? (ii) A particle limited to move along x-axis has the wave function \psi = ax between x = 0 and x = 1, \psi = 0 elsewhere. Find the probability that the particle can be found between x = 0\cdot 45 and x = 0\cdot 55. Find the expectation value <x> of the particles position x = 0 to x = 1.

Find the probability current density for the wave function \Psi(x,t)=\left[Ae^{ipx/\hbar}+Be^{-ipx/\hbar}\right]e^{-ip^2t/2m\hbar} Interpret the result physically.

A particle is described by the wave function \Psi(x)=\left(\frac{\pi}{2}\right)^{-1/4}e^{-ax^2/2}. Calculate \Delta x and \Delta p for the particle, and verify the uncertainty relation \Delta x\Delta p=\frac{\hbar}{2}.

Prove the following : (i) [L^2, L_x] = 0 (ii) [L_x, L_y] = i \hbar L_3

If the z-component of an electron spin is +\frac{\hbar}{2}, what is the probability that its component along a direction z' (forming an angle \theta with z-axis) is \frac{\hbar}{2} or -\frac{\hbar}{2}? What is the average value of spin along z'?

Prove the commutation relation for the angular momentum: [L^2,L_z]=0 Also show that (\vec{L}\times\vec{L})=i\hbar\vec{L}.