UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Find the values of the following : (i) \vec{L} \times \vec{L} (ii) L_+ L_- (iii) [L_z, L_+] where \vec{L} is the angular momentum operator. L_+ and L_- are raising and lowering operators respectively.

Obtain the normalized eigenvectors of \sigma_x and \sigma_y matrices.

An electron in a one-dimensional infinite potential well, defined by V(x)=0 for -a\leq x\leq a and V(x)=\infty otherwise, goes from n=4 to n=2 level and emits photon of frequency 3.43\times10^{14}\,\mathrm{Hz}. Calculate the width of the well. (Assume Plank's constant h=6.626\times10^{-34}\,\mathrm{J.S.} and mass of electron m=9.11\times10^{-31}\,\mathrm{kg})

The raising (J_+) and lowering (J_-) operators are defined by J_+=J_x+iJ_y and J_-=J_x-iJ_y respectively. Prove the following identities:

(i) [J_z,J_\pm]=\pm\hbar J_\pm

(ii) J_-J_+=J^2-J_z^2-\hbar J_z

Show that for a given principal quantum number n, there are n^2 possible states of the atom.

What is de Broglie concept of matter wave? Evaluate de Broglie wavelength of Helium that is accelerated through 300\mathrm{V}. (Given mass of proton = mass of neutron = 1.67\times10^{-27}\,kg)

What is the spin wave function (for s=\frac{1}{2}) if the spin component in the direction of unit vector \eta has a value of \frac{1}{2}\hbar?

Set up the Schrodinger's wave equation for one dimensional potential barrier and obtain the probability of tunnelling.

Show that E_n=\langle V\rangle in the stationary states of the hydrogen atom.

Calculate the Fermi energy of aluminium at absolute zero. The density of aluminium is 2\cdot 7 \times 10^3\text{ kg m}^{-3} and its atomic weight is 26\cdot 98\text{ kg (k mol)}^{-1}. Show that the electron gas in aluminium is strongly degenerate.

What do you understand by negative temperature? Write and explain various restrictions on a system for the concept of negative temperature to be meaningful.

A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heat-conducting piston which divides the cylinder in two parts. Initially, the piston is clamped in the centre, with one litre of air at 200\ \mathrm{K} and 2\ \mathrm{atm} pressure on one side and one litre of air at 300\ \mathrm{K} and 1\ \mathrm{atm} pressure on the other side. The piston is released and the system reaches equilibrium in pressure and temperature, with the piston at a new position. Compute the final pressure and temperature.

A compressor designed to compress air is used instead to compress helium. It is found that the compressor overheats. Explain this effect, assuming that the compression is approximately adiabatic and the starting pressure is same for both the gases. \left[\gamma_{\mathrm{He}}=\frac{5}{3},\gamma_{\mathrm{Air}}=\frac{7}{5}\right]

A gas of interacting atoms has an equation of state and heat capacity at constant volume given by the expressions p(T,V)=aT^{1/2}+bT^3+cV^{-2} C_v(T,V)=dT^{1/2}+eT^2V+fT^{1/2} where a through f are constants which are independent of T and V. Find the differential of the internal energy dU(T,V) in terms of dT and dV.

Two Carnot engines \text{C}_1 and \text{C}_2 operate in series, Engine \text{C}_1 absorbs heat at T and rejects heat to a sink at temperature 300\text{ K}. Engine \text{C}_2 absorbs \frac{1}{4}\text{th} of the heat rejected by engine \text{C}_1 and rejects heat to the sink at 200\text{ K}. If the work done in both the cases is the same, find the temperature T.

Assume that the Earth's atmosphere is pure nitrogen in thermodynamic equilibrium at a temperature of 300\ \mathrm{K}. Calculate the height above sea level at which the density of the atmosphere is one-half its sea level value. (Molecular weight of \mathrm{N}_2 is 28\mathrm{gm/mole})

A body of constant heat capacity C_p and a temperature T_i is put into contact with a reservoir at temperature T_f. Equilibrium between the body and the reservoir is established at constant pressure. Determine the total entropy change and prove that it is positive for either sign of \left[(T_f-T_i)/T_f\right]. Consider \left|\frac{T_f-T_i}{T_f}\right|<1.

Liquid {}^4\text{He} has normal boiling point at 4\cdot 2\text{ K}. It is observed that it boils at 1\cdot 2\text{ K} at a pressure of 1\text{ mm of Hg}. Calculate the average latent heat of vaporization of {}^4\text{He} in this temperature range. (\text{Given : Density of Hg} = 13\cdot 6 \times 10^3\text{ kg m}^{-3}).

One mole of gas obeys van der Waals equation of state. If its molar internal energy is given by u=cT-a/V (in which V is the molar volume, a is one of the constants in the equation of state and c is a constant), calculate the molar heat capacities C_v and C_p.

Estimate the temperature at which the root mean square speed of nitrogen molecules exceeds their most probable speed by 100\text{ ms}^{-1}.