PYQ Bank

Recall the Pauli matrix representation of two-state system of electron, \sigma_x, \sigma_y, \sigma_z, and show that they do not commute.

What does the uncertainty principle signify? Consider an electron in a box of size 10^{-10}\text{ m}. What will be its maximum uncertainty in momentum?

A typical atomic radius is about 5 \times 10^{-15}\,m and the energy of \beta-particle emitted from a nucleus is at most of the order of 1\,\mathrm{MeV}. Prove on the basis of uncertainty principle that the electrons are not present in nuclei.

An electron is confined to move between two rigid walls separated by 10^{-9}\,\mathrm{m}. Compute the de Broglie wavelengths representing the first three allowed energy states of the electron and the corresponding energies.

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.

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).

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}.

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}}.

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.

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

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.

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 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.

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

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.

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.

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.

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.

In free space, the electric field of electromagnetic wave is given as \vec{E}(x,t)=120\cos(\omega t-kx)\hat{y}\,\mathrm{V/m}. Find the average power crossing a circular area of radius one metre in the yz-plane.