UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Explain the phenomenon of penetration of a particle through a barrier whose height exceeds the total energy of the particle with necessary diagram.

An alpha particle is trapped in a nucleus of radius 1\cdot 4\text{ fm}. What is the probability that it will escape from the nucleus if its energy is 2\text{ MeV}? The potential barrier at the surface of the nucleus is 4\text{ MeV}. [Given : The mass of the alpha particle is 6\cdot 64 \times 10^{-27}\text{ kg}]

The raising (J_+) and lowering (J_-) operators are defined by J_+ = J_x + i J_y and J_- = J_x - i J_y. Show that– (i) [J_z, J_\pm] = \pm \hbar J_\pm (ii) J_+ J_- = J^2 - J_z^2 + \hbar J_z

Obtain the expressions for reflection coefficient (R) and transmission coefficient (T) for reflected waves and transmitted waves from an infinite thin barrier.

(i) Derive an expression for density of states for a free electron gas in one dimension. Hence, show its variation with energy for a one-dimensional metallic crystal. (ii) Where do you find the applications of free electron gas model?

Determine the ground state energy of an electron in an infinite potential well of width of 2\text{ \AA}.

Explain how the uncertainty in position is different from the uncertainty or inaccuracy of the measuring instruments.

V(x) = \begin{cases} 0, & x < -a \\\\ V, & -a < x < a \\\\ 0, & x > a \end{cases} For a potential with the boundary conditions V(x) = \begin{cases} 0, & x < -a \\\\ V, & -a < x < a \\\\ 0, & x > a \end{cases} solve the Schrödinger's equation in one dimension and find out the conditions for tunnelling.

State how for spin-half particles, the spin (\sigma) can be expressed by its three components \sigma_x, \sigma_y and \sigma_z.

g(E) = \frac{V}{\pi^2 \hbar^3 c^3} E^2 What is the density of states? For a relativistic particle of rest mass \mu, prove that the density of states in the extreme relativistic limit (E \gg \mu c^2) g(E) = \frac{V}{\pi^2 \hbar^3 c^3} E^2 where g(E) = density of states, V = volume of the system containing the particle, E = total energy, c = velocity of light and \hbar = Planck's constant.

By applying the Schrödinger's equation to the ground state of hydrogen atom, determine the zero-point energy.

A bullet of mass 0\cdot 025\text{ kg} is moving with a velocity of 600\text{ m/s}. The speed is measured with an accuracy of 0\cdot 02\%. Find out the uncertainty in x. Also, comment on the result.

A reversible heat engine operates with three reservoirs at 300\text{ K}, 400\text{ K} and 1200\text{ K}. It absorbs 1200\text{ kJ} energy as heat from the reservoir at 1200\text{ K} and delivers 400\text{ kJ} work. Determine the heat interactions with the other two reservoirs.

Consider a mixture of N_A molecules of a monatomic gas A and N_B molecules of a monatomic gas B. For this mixture, obtain the Helmholtz free energy and pressure. (The particle partition function for a monatomic gas is q = \left(\frac{2\pi m k T}{h^2}\right)^{\frac{3}{2}} V).

For silver, the specific heat at constant pressure in the range of 50\text{ K} to 100\text{ K} is given by : \text{C}_{\text{p}} = 0\cdot 076\text{ T} - 0\cdot 00026\text{ T}^2 - 0\cdot 15\text{ cal mol}^{-1}\text{deg}^{-1} If 2 moles of silver are heated from 50\text{ K} to 100\text{ K}, calculate the change in entropy.

Explain why, at equilibrium, the chemical potential of a component must be the same in all coexisting phases. Derive the equilibrium condition for a binary liquid-vapour system in terms of chemical potential.

One gram of water (1\text{ cm}^3) becomes 1671\text{ cm}^3 of steam when boiled at constant pressure of 1\text{ atm} (1\cdot 013 \times 10^5\text{ Pa}). The heat of vapourisation at this pressure is \text{L} = 2\cdot 256 \times 10^6\text{ J/kg}. Calculate :

(i) The work done by the water when it vapourizes, and

(ii) Increase in its internal energy.

Consider that an ideal non-interacting Fermi gas with internal energy '\text{U}' at temperature \text{T} is kept in a cubical box of volume \text{V}. Find the pressure for the gas in terms of \text{U} and \text{V}.

Discuss briefly the considerations which led Van der Waals to modify the gas equation. What are the critical constants of a gas ? Calculate the values of these constants in terms of the constants of the Van der Waals equation.

Consider \text{N} non-interacting ideal spinless particles (Bose gas) are occupying a volume \text{V}. Find out the temperature '\text{T}' below which B-E condensation takes place.