Search, filter, and study UPSC Civil Services & IFoS Physics past year questions with rigorous derivations, formulas, and step-by-step solutions.
2^2S_{y_2} level in H atom is 1058 \text{ MHz} above the 2^2P_{y_2} level. (i) What is this known as ? (ii) Express the above frequency in \text{cm}^{-1}. (iii) Calculate the energy difference between the above two levels in eV.
(i) Establish that hc = 1240 \text{ eV. nm} = 1240 \text{ MeV. fm} (ii) The energy levels of a hydrogen atom are given by E_n = (-1/n^2) \text{ Ryd,} where 1 \text{ Ryd} = hcR. Show that R = 1 \cdot 097 \times 10^7 \text{ m}^{-1}. What exactly is R ?
Explain fluorescence and phosphorescence in electronically excited molecules.
(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.
(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} .
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) 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.
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.
Using dimensional analysis, explain why the angular momentum of a particle cannot be \hbar^2.
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}})
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.
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 long solenoid of radius R and n turns per unit length carries a sinusoidal current I = I_0 \cos \omega t. Determine the magnitude of induced electric field (E) outside the solenoid.
A cylindrical conductor is carrying a current along its axis which is assumed to be in z-direction. The current is uniformly distributed throughout its cross-section. Show that the vector potential \vec{A} associated with the magnetic induction due to the current-carrying cylindrical conductor is independent of z.
A series R\text{-}L\text{-}C circuit is connected across a voltage source V = 100 \sin 300t. If R = 500\ \Omega, L = 1\text{ H} and C = 2\ \mu\text{F}, calculate the average power delivered to the circuit.
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.
Consider in the region 0 \le z \le 1\text{ m} an infinite slab made of a material with relative permeability, \mu_r = 3\cdot 5. If \vec{B} = (2y\hat{i} - 5x\hat{j}) \times 10^{-3}\text{ Wb/m}^2 within the slab, determine magnetisation \vec{M}.