Physics — UPSC Mains Optional PYQ (2024–2025)

(a) Explain how the uncertainty in position is different from the uncertainty or inaccuracy of the measuring instruments. [10M]

(b) Determine the ground-state energy of an electron in an infinite potential well of width 2 Å. [10M]

(c) Draw the normal Zeeman pattern for the 1F3 → 1D2 transition. [10M]

(d) In case of pure rotational states, if the temperature is doubled, calculate the rotational quantum number corresponding to maximum population density. [Assume that temperature is high.] [10M]

(e) The quantum numbers of two electrons in a two-valence-electron atom are n1 = 6, l1 = 3, s1 = 1/2 and n2 = 5, l2 = 1, s2 = 1/2. Assuming L–S coupling, find the possible values of L and J. [10M]

(a) What is the density of states? For a relativistic particle of rest mass μ, prove that in the extreme relativistic limit (E ≫ μc²) the density of states is g(E) = V⁄(π²ħ³c³) E², where g(E) is the density of states, V the volume of the system containing the particle, E the total energy, c the velocity of light and h Planck’s constant. [20M]

(b) Obtain the expressions for reflection coefficient (R) and transmission coefficient (T) for reflected waves and transmitted waves from an infinite thin barrier. [15M]

(c) For a potential V(x) = 0 (x < –a), V (–a < x < a) = V, and V(x) = 0 (x > a), solve the one-dimensional Schrödinger equation and find the conditions for tunnelling. [15M]

(a) Find the difference in frequencies of the Lyman-alpha line in hydrogen and deuterium atoms. [15M]

(b) The Stern–Gerlach experiment is a landmark experiment in quantum mechanics. Discuss the most important findings of this experiment. [15M]

(c(i)) From the pure rotational absorption spectra of a diatomic molecule (HF), the wave-number difference between consecutive rotational lines is Δν̄ = 4050 m⁻¹. Calculate (1) the rotational constant, (2) the moment of inertia, and (3) the bond length. [Given, M_H = 1 u, M_F = 19 u] [10M]

(c(ii)) The force constant of HCl molecule is 4.8 × 10⁵ dyne cm⁻¹. Calculate the wave numbers of Stokes and anti-Stokes lines when excited with radiation of wavelength 4358 Å. [Given, μ_HCl = 1.61 × 10⁻²⁴ g] [10M]

(a) State how, for spin-half particles, the spin (σ) can be expressed by its three components σ_x, σ_y and σ_z. [20M]

(b) By applying the Schrödinger equation to the ground state of hydrogen atom, determine the zero-point energy. [15M]

(c) Distinguish between fluorescence and phosphorescence. Explain the mechanisms responsible for these phenomena and discuss their applications in fields such as biochemistry and material science. [15M]

(a) Show that the energy of the triplet state (S = 1) is not equal to the energy of the singlet state (S = 0). [10M]

(b) ρ⁰ and K⁰ mesons both decay mostly to π⁺ and π⁻. Why is the mean lifetime of ρ⁰ equal to 10⁻²³ s, whereas that of K⁰ is 0.89 × 10⁻¹⁰ s? [10M]

(c) Find the radius of the interstitial sphere that can just fit into the void at the body centre of the fcc structure coordinated by the face-centred atoms. [10M]

(d) In the powder-diffraction pattern for lead with radiation of wavelength λ = 1.54 Å, the (220) Bragg reflection angle is θ = 32°. Find the atomic radius. [10M]

(e(i)) What are the differences in the electrical characteristics of FET (JFET) and MOSFET? [7M]

(e(ii)) How does an n-channel FET differ from a p-channel FET? [3M]

(a) The total binding energies of ¹⁵₈O, ¹⁶₈O and ¹⁷₈O are 111.96 MeV, 127.62 MeV and 131.76 MeV respectively. Determine the energy gap between the 1p₁∕₂ and 1d₅∕₂ neutron shells for a nuclide whose mass number is close to 16. [15M]

(b) State the basic assumption of the single-particle shell model. How do the centrifugal and spin-orbit terms remove the degeneracy of a three-dimensional spherical harmonic oscillator? [20M]

(c) Explain the members of the leptonic family. What is leptonic number conservation? Based on this law, state whether the following reactions are possible or not: (i) π⁻ → μ⁻ + ν̄_τ (ii) n → p⁺ + e⁻ + ν̄_e [15M]

(a) What is the minimum energy required to break a ²He⁴ nucleus into free protons and neutrons? [Given, m_H = 1.007825 amu, m_n = 1.008665 amu, m_e = 0.00055 amu and m_He = 4.002603 amu] [15M]

(b(i)) Consider a uranium nucleus (₉₂U²³⁶) breaking up spontaneously into two equal parts. Estimate the reduction of electrostatic energy of the nucleus assuming a uniform charge distribution. [Assume the nuclear radius R = 1.2 × 10⁻¹³ A¹∕³ cm] [15M]

(b(ii)) Is it possible for a photon to transfer all its energy to a free electron? Give reasons. [5M]

(a) A particle of mass m kg having an initial velocity V0 is subjected to a retarding force proportional to its instantaneous velocity. Obtain the expression for the velocity and position of the particle as a function of time. [10M]

(b) Show that the kinetic energy of a system of n particles is given by T = 1/2 M VCM^2 + 1/2 Σ (i = 1 to n) mi Vi'^2 where M is the total mass, VCM is the velocity of the centre of mass, Vi' is the velocity of the particles about the centre of mass and mi is the mass of the ith particle. [10M]

(c) A charged π-meson with rest mass of 273 me at rest decays into a neutrino and a μ-meson of rest mass 207 me. Find the kinetic energy of the μ-meson and the energy of the neutrino. (me is the rest mass of the electron) [10M]

(d) The intensity at the central maximum observed on a screen in a double-slit experiment is 2 × 10^−3 W m^−2. If the path difference between interfering waves reaching a point on the screen is λ⁄6, where λ is the wavelength of the light used in the experiment, determine the intensity at that point. [10M]

(e) A telescope has an objective lens of diameter 10 cm. Determine whether this telescope can resolve two stars having an angular separation of 2·4 seconds of arc. (Assume the wavelength of starlight as 550 nm) [10M]

(a) A particle limited to the x-axis has the wave function φ(x) = bx² between x = 0 and x = 2; φ(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1·0 and x = 1·5. (ii) Find the expectation value < x > of the particle position. [10M]

(b) Show that the square of the orbital angular momentum operator (L²) commutes with any of the components of angular momentum operator L. Is it possible to measure L², Lx, Ly and Lz simultaneously? Give reasons for your answer. [10M]

(c) How is Rydberg constant related to emission wavelength of hydrogen spectrum? [10M]

(d) Explain how the hydrogen spectrum is used for imaging the universe. [10M]

(e) Find the energy of the particle of mass m moving in a potential field V(x) = 2ħ²b²x² / m for which the time-independent wave function is ψ(x) = exp(–bx²). Here b is a constant. [10M]

(a(i)) Briefly discuss the Kepler's laws of planetary motion. [5M]

(a(ii)) Show that the escape velocity Ve on the surface of the Earth is given by Ve = √2 g R, where g = 9·8 m s^−2 and R is the radius of the Earth. [5M]

(a(iii)) Two satellites A and B of same mass are orbiting the Earth at altitudes R and 5 R, respectively, where R is the radius of the Earth. Assuming their orbits to be circular, calculate the ratios of their kinetic and potential energies. [5M]

(b) Show that the angular momentum of a rigid body consisting of n particles of masses mi (i = 1, 2, 3, …, n) rotating with an instantaneous angular velocity ω about an axis passing through the origin O of the coordinate system OXYZ is given by L = I · ω, where I is known as the inertia tensor. [20M]

(c) A weakly damped harmonic oscillator consisting of a spring–mass system has the following parameters: Mass m = 0·25 kg, Spring constant k = 100 N m^−1, Damping coefficient γ = 1 N s m^−1. A periodic force F = 5 cos ωt (newton) is applied to the system. Determine (i) the amplitude of the oscillator at resonance and (ii) the Q-value of the oscillator. [15M]

(a) Prove that (i) [L², Lz] = 0; (ii) [Lz, L+] = ħL+; (iii) [L+, L–] = 2ħLz; (iv) L+ Ly = L² – Lz² + ħLz; where ħ = h / 2π. [20M]

(b) For the ground-state wave function ψ₀(x) = (mω / πħ)^{1/4} exp(–mωx² / 2ħ) of a harmonic oscillator: (i) At which point is the probability density maximum? (ii) What is the value of the maximum probability density? [15M]

(c) (i) Assuming the nucleus to present a one-dimensional infinite rigid-wall potential of length 10⁻¹⁵ m, estimate the minimum kinetic energy of the electron. (ii) Estimate the minimum kinetic energy of a neutron bound within such a nucleus. Can an electron be confined in a nucleus? Explain. [15M]

(a(i)) Explain the phenomenon of double refraction. What are positive and negative crystals? Give their examples. [5M]

(a(ii)) What do you understand by optical activity? A linearly polarized light is propagating along the optic axis of a quartz crystal of thickness 0·2 cm. If the difference in the refractive indices corresponding to right circularly polarized and left circularly polarized beams is 7 × 10^−5 and the wavelength of the light is 0·5 μm, calculate the angle of polarization. [10M]

(b(i)) What do you understand by attenuation in optical fibres? What are the factors responsible for the attenuation? [5M]

(b(ii)) Consider a 10 mW laser beam passing through a 50 km fibre link of attenuation 0·5 dB km^−1. Calculate the power of the laser at the end of the link. [10M]

(c(i)) State and explain Hooke's law of elasticity. Briefly discuss the features of the stress–strain diagram for the behaviour of a wire undergoing increasing stress. [10M]

(c(ii)) Explain the Poiseuille's equation for the rate of flow of a liquid through a capillary tube. From this, show that if two capillary tubes of radii r1 and r2 having lengths l1 and l2, respectively, are connected in series, the rate of flow of the liquid is given by Q = (π P)/(8 η) ( l1/r1^4 + l2/r2^4 )^−1, where P is the pressure across the arrangement and η is the coefficient of viscosity of the liquid. [10M]

(a) How do Stokes lines appear in Raman spectrum as per classical and quantum theory of Raman effect? [20M]

(b) What is Lamb shift in the fine structure of hydrogen spectrum? Discuss its theory based upon second quantization. [15M]

(c) Describe Electron Paramagnetic Resonance. Highlight its differences with NMR and discuss its applications. [15M]

(a(i)) Write down the system (transfer) matrix for a combination of two thin lenses in paraxial approximation. Hence obtain the focal length of the combination and the positions of unit planes. [10M]

(a(ii)) Consider a thin-lens combination of two convex lenses of focal lengths f1 = +10 cm and f2 = +20 cm, respectively, kept separated by 25 cm. Determine the focal length of the combination and the positions of unit planes. [10M]

(b) The diameter of the central zone of a zone plate is 2·4 mm. If a point source of light of wavelength 600 nm is placed at a distance of 5·0 m from the zone plate, calculate the position of the first image. [10M]

(c(i)) Consider three inertial frames O, O′ and O″. Frame O′ moves with velocity V relative to O and frame O″ moves with velocity V′ relative to O′, both velocities being along the same direction. Write down the transformation relations connecting the coordinates x, y, z, t with x′, y′, z′, t′ and those connecting x′, y′, z′, t′ with x″, y″, z″, t″. (Assume the velocities are along the x-axis.) [10M]

(i) Using free electron theory of metals, calculate the Fermi energy level of sodium atom at absolute zero. Assume sodium has one free electron per atom and its density is 0·97 gm cm⁻³. [10M]

(ii) Draw the energy level diagram and give mathematical expressions for (I) En of an electron confined in a one-dimensional box, and (II) a linear harmonic oscillator. Make a qualitative comparison of these two cases. [10M]