UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Solve the Schrodinger equation for a step potential and calculate the transmission and reflection coefficient for the case when the kinetic energy of the particle E_0 is greater than the potential energy V (i.e., E_0>V).

Consider a two-electron system in ground state with orbital quantum number zero (l = 0). Use Pauli's exclusion principle to obtain orientations of the two electrons.

Explain the phenomenon of Zeeman effect of atomic physics.

Write down the matrix representation of the three Pauli matrices \sigma_x, \sigma_y and \sigma_z. Prove that these matrices satisfy the following identities:

(i) [\sigma_x,\sigma_y]=2i\sigma_z

(ii) [\sigma^2\cdot\sigma_x]=0

(iii) (\vec{\sigma}\cdot\vec{A})(\vec{\sigma}\cdot\vec{B})=\vec{A}\cdot\vec{B}+i\vec{\sigma}\cdot(\vec{A}\times\vec{B}) if \vec{A} and \vec{B} commute with Pauli matrices.

In a nuclear experiment, beams of electrons and protons are moving with velocities c / 10 and c / 20 respectively. Calculate the de Broglie wavelengths of both the particles in metre (c is the speed of light = 3 \times 10^8\text{ m / s}).

Calculate the density of states for an electron moving freely inside a metal with the help of quantum mechanical Schrodinger's equation for free particle in a box.

The molecules of a gas obeying Maxwell-Boltzmann distribution move with an average speed of 450\ \mathrm{m\ s^{-1}}. If the coefficient of viscosity of the gas \eta is 16.6\times10^{-6}\ \mathrm{N\,s\,m^{-2}}, density of the gas \rho is 1.25\ \mathrm{kg\ m^{-3}} and number density is 2.7\times10^{25}\ \mathrm{m^{-3}}, calculate the mean free path and diameter of the gas molecules.

The molecules of a gas obey Maxwell-Boltzmann distribution. Calculate the fraction of molecules of the gas within 1\% of the most probable speed at STP. Interpret your result.

Consider a system of N particles and a phase space consisting of only two states with energies 0 and \varepsilon(>0). Obtain the expressions for the partition function and the internal energy of the system, if it obeys M-B statistics.

The viscosity in a liquid arises due to friction between adjacent layers. What causes viscosity in a gas? Explain.

What do you understand by the term 'phase transition'? Using Clausius-Clapeyron equation, show that for first-order phase transitions, vapour pressure decreases exponentially with temperature. You can assume that the vapour behaves like an ideal gas and latent heat remains constant with temperature.

Write down van der Waals' equation of state for n moles of a gas and calculate the temperature at which 5 moles of the gas at 5\ \mathrm{atm} pressure will occupy a volume of 20 litres. Given, R=8.31\times10^7\ \mathrm{erg\,mol^{-1}\,K^{-1}}, a=1.34\times10^{12}\ \mathrm{dyne\,cm^4\,mol^{-2}}, b=31.2\ \mathrm{cm^3\,mol^{-1}} and 1\ \mathrm{atm}=1.013\times10^6\ \mathrm{dyne\,cm^{-2}}.

Write down the salient features of the Einstein's theory of lattice heat capacity. Further write down the expression for specific heat in Einstein's theory and explain its high and low temperature limits.

m gram of water at temperature T_1 is isobarically and adiabatically mixed with an equal mass of water at temperature T_2. Show that the change in entropy is given by \Delta S=2mC_p\ln\left(\dfrac{T_{\mathrm{av}}}{T_{\mathrm{geo}}}\right), where T_{\mathrm{av}}=\dfrac{T_1+T_2}{2} and T_{\mathrm{geo}}=\sqrt{T_1T_2}.

A system of non-interacting fermions enclosed in a volume, V, is at T = 0^\circ\text{K}. Find an expression for the internal energy, U, of the system.

Derive the expression for the specific heat of a solid based on Einstein's theory. Obtain the limiting form of the specific heat at very low temperature.

Write down the expression for the Bose-Einstein distribution function and explain the meaning of the symbols used.

A quasistatic isothermal and quasistatic adiabatic intersect in a p-v diagram at more than one point as shown in diagram. Does it imply violation of 2nd law of thermodynamics ? Give reasons for your answer.

The mean speed of the molecules of an ideal monatomic gas when it contracts (or expands) adiabatically depends on the pressure according to the law v = K p^n, where K is a constant. Find n.

(i) Show that the work done per unit volume of a ferromagnetic substance per one complete cycle of hysteresis is given by W = \oint \vec{H} \cdot d\vec{B} (ii) Further, show that the value of W is equal to the area of B-H loop. Which material is preferred for the core of a transformer and why ?