UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Give an account of Heisenberg's Uncertainty principle. Outline an idealised experiment to bring out its significance.

Calculate the most probable value of 'r' for an electron in the ground state of the hydrogen atom.

Find the de Broglie wavelength of a neutron moving with a kinetic energy of 500\text{ eV}. (1\text{ eV} = 1\cdot 602 \times 10^{-19}\text{ J})

Determine the values of the total angular momentum for a 3d electron.

Deduce the commutation relations between the components of angular momentum operator \mathbf{L} [L_x, L_y] = i \hbar L_z [L_y, L_z] = i \hbar L_x [L_z, L_x] = i \hbar L_y using the commutation relations [x, p_x] = [y, p_y] = [z, p_z] = i \hbar

Normalized wave function of a particle is given: \psi(x)=N\exp\left(-\frac{x^2}{2a^2}+ikx\right). Find the expectation value of position.

What is Zeeman effect? How can it be understood on the basis of quantum mechanics?

Write the time independent Schrödinger equation for a bouncing ball.

Solve the Schrödinger equation for a particle in a three-dimensional rectangular potential barrier. Explain the terms degenerate and non-degenerate states in this context.

A particle trapped in an infinitely deep square well of width a has a wave function \psi=\left(\frac{2}{a}\right)^{1/2}\sin\left(\frac{\pi x}{a}\right). The walls are suddenly separated by infinite distance. Find the probability of the particle having momentum between p and p+dp.

Establish that: hc = 1240\ \mathrm{eV\,nm} = 1240\ \mathrm{MeV\,fm}

Derive an expression for the total number of particles N in an ideal Bose gas at any temperature T. Hence, obtain the relation N_0 = N \left[ 1 - \left( \frac{T}{T_0} \right)^{\frac{3}{2}} \right] for the number of particles N_0 in the ground state at T < T_0, the Bose-Einstein condensation temperature, and draw N_0 versus T curve.

The vapour pressure of an organic substance is 50\times10^3\ \mathrm{Pa} at 40^{\circ}\mathrm{C}. Its normal boiling point is 80^{\circ}\mathrm{C}. If the substance in vapour phase can be treated like an ideal gas, find the latent heat of vaporization of the substance.

For a Van der Waals gas, write down the equation of state. Determine the coefficient of critical expansion \beta.

A Van der Waals gas undergoes Joule-Kelvin expansion with a pressure drop of 50\ \mathrm{atm}. If its initial temperature is 300^{\circ}\mathrm{K}, determine its final temperature. (Given Van der Waals constant a=0.136\ \mathrm{Pa\,m^6\,mol^{-1}}, b=36.5\times10^{-6}\ \mathrm{m^3\,mol^{-1}}, C_p=30\ \mathrm{J\,K^{-1}\,mol^{-1}}, R=8.3\ \mathrm{J\,K^{-1}\,mol^{-1}}.)

Discuss the consequence following from Joule's free expansion experiments in the context of the internal energy of an ideal gas.

Start from the equation T dS = C_p dT - T \left( \frac{\partial V}{\partial T} \right)_P dP and get the relation \left( \frac{\partial C_p}{\partial P} \right)_T = -T \left( \frac{\partial^2 V}{\partial T^2} \right)_P Hence, show that the third law of thermodynamics requires for the coefficient of thermal expansion of any substance to vanish at T = 0.

Plot the Fermi distribution function versus energy at temperatures T = 0 and T > 0. Explain the nature of the former curve on the basis of the Pauli principle.

Using Planck's radiation law, deduce Wien's displacement law. How does this law enable one to estimate the surface temperature of the Sun or a star?

What are the characteristic features of Rayleigh scattering? A very thin monochromatic beam of light is incident on a particle. Suggest a simple experimental method to ascertain whether the scattering by the particle is of Rayleigh type.