UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Describe normal and anomalous Zeeman effect. Explain how it lifts the degeneracy in hydrogen atom.

Define angular momentum of a particle and find out the three components of the angular momentum operator \hat{L} in Cartesian coordinates. Show that \hat{L}^2=-\hbar^2\left[r^2\nabla^2-\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)\right] Prove that the operator \hat{L}^2 can also be expressed as \hat{L}^2=-\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] in spherical polar coordinates (r,\theta,\phi).

Estimate the size of hydrogen atom and the ground state energy from the uncertainty principle.

For a free quantum mechanical particle under the influence of a one-dimensional potential, show that the energy is quantized in discrete fashion. How do these energy values differ from those of a linear harmonic oscillator?

State and express mathematically the three uncertainty principles of Heisenberg. Highlight the physical significance of these principles in the development of Quantum Mechanics.

Define mathematically the Bohr radius of a hydrogen atom and show that the binding energy at state n of this atom can be given by E_n=-\frac{1}{2}\frac{Ze^2}{(a/Z)4n^2\pi\epsilon_0} where Z is the atomic number of H atom. Calculate the numerical values of a and E_1 of H atom.

How do you define density of states? Show that the density of states with wave vector less than \vec{k} in a three-dimensional cubic box of volume V can be given by D(\omega)=\frac{V}{2\pi^2}k^2\left(\frac{dk}{d\omega}\right) in the frequency spectrum between \omega and \omega+d\omega. Here, assume that the number of modes per unit range of k is L/(2\pi), L being the length of each side of the cubic box.

Show that the mass and linear momentum of a quantum mechanical particle can be given by m=h/(\lambda v) and p=h/\lambda, respectively, where h, \lambda and v are Planck’s constant, wavelength, and velocity of the particle, respectively. Comment on the wave-particle duality from these relations.

Define Pauli spin matrices \sigma_x, \sigma_y and \sigma_z. Using these definitions, prove the following:

(i) \sigma_x^2=\sigma_y^2=\sigma_z^2=1

(ii) \sigma_x\sigma_y=i\sigma_z;\ \sigma_z\sigma_x=i\sigma_y;\ \sigma_y\sigma_z=i\sigma_x

If the partition function for a perfect gas is given by Z=\frac{V}{h^3}(2\pi m k T)^{3/2}, calculate (i) average kinetic energy per molecule and (ii) specific heat of the gas.

Explain the effect of pressure on the melting and boiling points of a substance using Clapeyron's latent heat equation. Calculate under what pressure, water will boil at 120^\circ\mathrm{C}, if the change in specific volume when 1 gram of water is converted into steam is 1676\ \mathrm{cm^3}. Latent heat of steam =540\ \mathrm{cal/g}, 1 atmospheric pressure =10^6\ \mathrm{dynes/cm^2}.

What are the conditions for the change in temperature of a van der Waals gas passing through a porous plug? Prove that the ideal gas passing through the porous plug does not show any change in temperature.

What is Carnot's theorem? Prove that Carnot's reversible engine is the most efficient one and no other engine can be more efficient than Carnot's engine.

A gas has only two particles, a and b. With the help of a diagram, show that how these two particles can be arranged in the three quantum series 1, 2, 3 using (i) Maxwell-Boltzmann, (ii) Fermi-Dirac, and (iii) Bose-Einstein statistics.

Derive the expression for the average energy of a quantum oscillation of frequency \nu. Assume Fermi-Dirac distribution and E-E_F>2, where E_F is the Fermi level.

What is Gibbs' phase rule? Find the values of degrees of freedom when

(i) only the liquid \mathrm{CO_2} is in equilibrium with the gaseous \mathrm{CO_2}.

(ii) water is in the vapour-liquid saturation region.

(iii) water is in a single-phase region.

(iv) water is at the triple point.

Einstein's molar specific heat capacity of a solid is given by C_V = 3R\left(\frac{\theta_E}{T}\right)^2\frac{e^{\theta_E/T}}{\left(e^{\theta_E/T}-1\right)^2}, where \theta_E=\frac{\hbar\omega}{k_B}. Obtain the expressions for the cases:

(i) when T \gg \theta_E

(ii) when T \ll \theta_E What is the discrepancy of Einstein model to explain the variation of specific heat capacities of solids with the temperature? The molar specific heat capacity of a solid at constant volume is 2.77\ \mathrm{J\,K^{-1}} at 36.8\ \mathrm{K}. Determine the Debye temperature of the solid.

Two conducting planes, intersecting at right-angles to each other, are kept at a potential \phi_0. Calculate the potential at a point in space if the total charge on a plane of area \alpha be Q.

Why do we prefer to work with a critically damped ballistic galvanometer in a laboratory? What is external critical damping resistance?

Three cells are connected in parallel with similar poles connected together with wires having negligible resistance. The emfs of the cells are 2, 1 and 4 volts respectively and the corresponding internal resistances are 4, 3 and 2 ohms. Calculate the current flowing through the 4 V cell.