UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Explain fluorescence and phosphorescence in electronically excited molecules.

Explain Born – Oppenheimer approximation. Discuss the intensity distribution in the vibrational electronic spectra of a diatomic molecule on the basis of Franck – Condon principle.

Discuss the vibrational spectra of a diatomic molecule treating it as a harmonic oscillator as well as an anharmonic oscillator and compare them.

Using dimensional analysis, explain why the angular momentum of a particle cannot be \hbar^2.

Show that the time-dependent part of all the solutions of the Schrödinger equation in one-dimension has the structure \phi(t) = \exp (- i E t / h), provided the potential is not an explicit function of time.

(i) Explain spin-orbit coupling of an atomic electron. (ii) Show that the 2p state in the H atom splits up into two substates due to spin-orbit coupling. (iii) Calculate the energy of separation in eV, resulting from the spin-orbit coupling when the magnetic field experienced by the electron is 0 \cdot 4 \text{ T}.

(i) Consider a positron in a box. If the energy released is 60 \text{ eV} when it jumps from the third excited state to the ground state, show that the width of the potential is nearly 0 \cdot 3 \text{ nm}. (ii) Prove that the most probable distance of an electron from the proton (in the hydrogen atom) is the Bohr radius of the hydrogen atom. Consider only the ground state.

Show that the Pauli Spin Matrices obey the following relations : (i) \text{Tr} (\sigma_x) = \text{Tr} (\sigma_y) (ii) \det (\sigma_y) = \det (\sigma_z) (iii) The eigenvalues of \sigma_z and \sigma_x are the same. (iv) Write down the y-component of the spin angular momentum matrix corresponding to an antineutrino.

(i) The quantum mechanical probability distribution function of an electron in the ground state of the hydrogen atom is P(r) = N r^2 \exp (-2br). Using the result \int_0^\infty P(r) \, dr = 1, deduce that N is proportional to b^3. (ii) Prove that the value of 40 \, k_B T at T = 300 \text{ K} is nearly 1 \text{ eV}. Hence determine the Fermi temperature of a metal whose Fermi energy is 9 \cdot 4 \text{ eV}. (iii) Show that the Fermi velocity is related to the Fermi energy of electrons through the relation \frac{v_F}{c} = 1 \cdot 98 \left( \frac{E_F}{1 \text{ MeV}} \right)^{1/2} .

(i) Consider a particle in a three-dimensional box. Derive an expression for g(E), the density of states. (ii) Show that \frac{g(p)}{g(E)} = \frac{dE}{dp} , where g(p) is the density of states in the momentum space. Deduce that g(p) is proportional to p^2 for a free non-relativistic particle.

What are the limitations of Einstein's theory of specific heat of solids when compared with experiments at low temperature? Outline the assumptions made in Debye's theory and show that the specific heat at low temperature follows C_v \sim T^3 law. What is the significance of Debye's temperature, T_D?

Derive an expression for the specific heat of a solid on the basis of Debye's model. Show that it converges to Dulong and Petit's law at high temperatures.

Energy distribution for n_i particles in classical statistical mechanics is given by n_i = g_i e^{-\alpha -\beta \varepsilon_i} where \alpha and \beta are constants. g_i is the single particle states in the i\text{ th} level. Using equipartition theorem, show that correct thermodynamic interpretation is \beta = \frac{1}{kT} (Use \int_0^\infty e^{-x} x^{1/2} \, dx = \frac{\sqrt{\pi}}{2} and \int_0^\infty e^{-\beta x} x^{3/2} \, dx = \frac{3\sqrt{\pi}}{4\beta^{5/2}})

Define entropy. How is it related to disorder? Hence, derive the Boltzmann relation S = k \log \Omega, where \Omega is the probability and k is the Boltzmann constant. Show that for any type of process, involving a closed system \Delta S \ge \frac{\Delta Q}{T} where the equality sign applies for internally reversible processes and the inequality for internally irreversible processes.

State Gibbs phase rule. Show that for a 1-component closed thermodynamic system having two phases, the condition for equilibrium between the phases is that their specific Gibbs functions are equal.

A thin dielectric cylindrical rod of cross-section A is situated along z-axis from z = 0 to z = L. The polarisation of the rod is along its length and it is given by \vec{P} = (2z^2 + 5)\hat{z}. Calculate bound volume charge density at each end of the rod.

Using Planck's radiation formula, deduce Wien's displacement law.

In a non-charged current-free dielectric, \rho = 0 and \vec{J} = 0. Show that in this medium, electric (\vec{E}) and magnetic (\vec{H}) fields satisfy three-dimensional wave equations \nabla^2 \vec{E} = \varepsilon\mu \frac{\partial^2 \vec{E}}{\partial t^2} \quad \text{and} \quad \nabla^2 \vec{H} = \varepsilon\mu \frac{\partial^2 \vec{H}}{\partial t^2} Using Poynting theorem of electromagnetic theory, describe the significance of the vector \vec{P} = (\vec{E} \times \vec{H}) and the scalar u = \frac{1}{2} [\vec{B} \cdot \vec{H} + \vec{D} \cdot \vec{E}]

A plane-polarised electromagnetic wave is incident on the interface of two dielectrics having dielectric permittivity \varepsilon_1 and \varepsilon_2. Assume that the electric vector \vec{E} lies in the plane of incidence. Using the boundary conditions at the interface, obtain the expressions for the amplitude reflection coefficient (r_{11}) and the amplitude transmission constant (t_{11}). Using the components of the Poynting vector \vec{E} \times \vec{H} associated with the reflected and transmitted waves, obtain the expressions for reflection and transmission coefficients R_{11} and T_{11} respectively. Under what condition, r_{11} = 0 and t_{11} = 1?

Consider the incidence of a plane-polarised electromagnetic wave at the interface of two media having dielectric permittivity and magnetic permeability (\varepsilon_1, \mu_1) and (\varepsilon_2, \mu_2) respectively. The interface is chosen to be x = 0 plane. \vec{K}_1, \vec{K}_2 and \vec{K}_3 represent the propagator vectors associated with the incident, refracted and reflected waves respectively. Using the boundary conditions on them, establish the Snell's laws of refraction.