UPSC Physics Optional PYQs (CSE & IFoS)PYQs

(i) State the quantum numbers I_z, Y and S for the uds quarks and antiquarks. What combination of these leads to the formation of (a) proton and (b) neutron ?

Describe how parity violation was experimentally detected in {}^{60}\text{Co} \beta-decay. How was the observed asymmetry in the distribution of emitted electrons explained ?

(i) How many lines occur in the multiplet arising from {}^{2}\text{P}_{3/2} \rightarrow {^{2}\text{S}_{1/2}} and {}^{2}\text{P}_{1/2} \rightarrow {^{2}\text{S}_{1/2}} transitions of alkali metal atoms placed in a weak magnetic field, why ? (ii) The wavefunction of a particle confined in a cube of volume L^3 is given by \Psi (x,y,z) = \left( \frac{2}{L} \right)^{3/2} \sin \frac{\pi x}{L} \sin \frac{\pi y}{L} \sin \frac{\pi z}{L} Calculate the average values of p_x and p_x^2 in the region 0 < x < L.

Obtain an expression for the vibration-rotation energy levels of a diatomic molecule in a given electronic state. The wave numbers of the vibrational transitions occuring in HF, HCl and HI molecules are 4141 \cdot 3 \text{ cm}^{-1}, 2988 \cdot 9 \text{ cm}^{-1} and 2309 \cdot 5 \text{ cm}^{-1} respectively. Compare the force constants of these three molecules.

(i) Explain the molecular phenomenon of spontaneous emission between two electronic states of the same multiplicity and differentiate it form that of different multiplicity. (ii) The constants B and V_0 for a KCl molecule have values 1 \cdot 43 \times 10^{-5} \text{ eV} 8 \cdot 4 \times 10^{12} \text{ s}^{-1} respectively. Determine the number of rotational levels between the vibrational levels v = 0 and v = 1.

Describe Stern-Gerlach experiment and discuss its implications.

(i) Write the commutation relations for the position variable x and the momentum components p_x, p_y and p_z. Explain the physical significance of these relations. (ii) Calculate the de Broglie wavelength of an electron moving with a kinetic energy of 1 MeV.

Discuss WKB approximation and apply the same to determine the transition probability for leakage through a potential barrier.

Set up the time-independent Schrodinger equation for an electron moving in Coulomb field, V(r) = \frac{Ze^2}{4 \pi \varepsilon_0 r}, in polar coordinates. Solve the radial equation to get the energy eigen values.

(i) The ground state wavefunction of a linear simple harmonic oscillator is \psi = A \exp \left( - \frac{\alpha^2 x^2}{2} \right) Calculate the constant A and the average values of x^2 and x. Given that \int_0^\infty e^{-x^2} = \frac{\pi^{1/2}}{2} (ii) How can the pure rotation spectrum of \text{H}_2 molecule be observed ? If the bond length of \text{H}_2 molecule is 0 \cdot 07417 \text{ nm}, what would be the spacing of lines in its spectrum ?

Consider a particle of mass m in an infinite one dimensional potential well of width a. The particle is found in the state given by \psi (x) = c \left[ \sin \frac{\pi x}{a} + \frac{1}{2} \sin \frac{2 \pi x}{a} \right] (i) Calculate c. (ii) If a measurement of energy is made, what are the possible results and what are the probabilities for each one of them ?

(c) Discuss the differences in the assumptions underlying Einstein and Debye theories of specific heat C_v. Give schematic plots of C_v versus reduced temperature for these theories and elucidate the differences therein. Elaborate the meaning of the “law of corresponding states” for these plots.

8. (a) Starting from the expression N = \sum_k <n_k> where <n_k> is the average number of particles in the k^{\text{th}} quantum state, derive an expression for the average number of particles in the ground state of an ideal Bose gas.

(b) Utilize the above expression to elaborate the concept of the Bose-Einstein condensation and discuss that the phenomenon explains qualitatively the properties in the low-temperature phase of liquid {}^4\text{He}.

(d) Consider the expression C_P - C_V = - T \left( \frac{\partial V}{\partial T} \right)_P^2 \left( \frac{\partial P}{\partial V} \right)_T and give reasoning regarding the values of T when C_P = C_V for water. Also, evaluate C_P - C_V for a vander Waals gas to elaborate that its value is larger for any real gas as compared to an ideal gas.

(c) Show that the chemical potential of a system is an intensive quantity and is a function of temperature and pressure only.

7. (a) Write down the macroscopic form of the Maxwell's equations in any isotropic (but inhomogeneous) medium and define the symbols appearing therein. Convert these equations in the integral forms to highlight the laws represented by these equations.

(c) Use the Planck formula for the blackbody radiation u(\omega, T) = \frac{\hbar^2}{\pi^2 c^3} \frac{\omega^3}{\exp(\beta \hbar \omega) - 1} with \beta = \frac{1}{k_B T} to derive Wien's law, Rayleigh-Jeans law and Stefan-Boltzmann law.

(b) Describe physical significance of the displacement current considering the example of current flow through a capacitor.

(b) Show that the electric and magnetic field vectors, \vec{E} and \vec{B}, in plane electromagnetic waves are mutually perpendicular in a plane normal to the direction of propagation. How are phases of \vec{E} and \vec{B} related to each other ?