UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Explain the salient features of nuclear shell model. What are the magic numbers? Write the experimental facts in support of shell model of nuclei.

What are the properties of the particles made up of the following quarks? Also write the names of particles :

(i) u\bar{d}

(ii) \bar{u}d

(iii) dds

(iv) uss

By applying the conservation laws, state about the following decay equations, whether they are allowed or not allowed :

(i) \Sigma^+ \rightarrow p + \pi^0

(ii) \Sigma^0 \rightarrow \Lambda^0 + \gamma

(iii) \Lambda^0 \rightarrow n + \pi^0

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}.

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

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

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?

Show that the phase velocity v_p for a particle with rest mass m_0 is always greater than the velocity of light and that v_p is a function of wavelength.

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.

Consider a particle whose wave function is given by \psi(x) = A e^{-\alpha x^2}.

(i) What is the value of A if this wave function is normalized?

(ii) Calculate the expectation value of x for this particle.

For Pauli's matrices, prove that (i) [\sigma_x, \sigma_y] = 2i\sigma_z and (ii) \sigma_x \sigma_y \sigma_z = i.

An electron and a photon each has a wavelength of 2~\text{\AA}. Calculate (i) their momenta and (ii) the ratio of their kinetic energies.

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.

If the Hamiltonian of a system be H = (p_x^2 / 2m) + V(x), obtain the value of the commutator [x, H]. Hence find the uncertainty product (\Delta x)(\Delta H).

Obtain an expression for the specific heat capacity of a solid on the basis of Einstein's theory. How far do the results from this theory agree with experimental data ?

Consider a quantum – mechanical gas of non-interacting spin zero bosons, each of mass m which are free to move within volume V. (i) Find the energy and heat capacity in the very low temperature region. Discuss why it is appropriate at low temperatures to put chemical potential equal to zero. (ii) Show how the calculation is modified for a photon (mass = 0) gas. Prove that the energy is proportional to \text{T}^4.

The equation of state of a dilute gas at very high temperature is described by \frac{\text{PV}}{\text{kT}} = 1 + \frac{\text{B(T)}}{\text{V}}, where V is the volume per particle and B(T) is a negative quantity. One can conclude that this is a property of a Van der Waal's gas. Explain why it is a property of Van der Waal's gas.