UPSC Physics 2021 — Paper II

Calculate numerical values for all five sub-parts with systematic derivations. Allocate approximately 20% time each to parts (a), (b), (c), (d), and (e) as they carry equal marks. Begin each sub-part by stating the relevant formula, show substitution with proper units, and conclude with the final numerical answer rounded as specified. For part (a), explicitly state the resolution criterion; for (d), show both Larmor frequency and wave number splitting.

• Part (a): Apply the criterion that Zeeman splitting must exceed spectrometer resolution; use Δλ = (eλ²B)/(4πmₑc²) or equivalent relation with Δλ = 0.010 nm to find B_min ≈ 0.43 T → round to 1 T (nearest high integer)
• Part (b): Calculate ⟨V⟩ = ⟨-e²/r⟩ using ψ₁₀₀; evaluate ∫ψ*(-e²/r)ψ d³r = -e²/πa₀³ ∫e^(-2r/a₀)(1/r)4πr²dr = -e²/a₀; final answer -27.2 eV or -2e²/a₀
• Part (c): Apply WKB or exact formula for rectangular barrier: T ≈ exp[-2κL] where κ = √[2m(V₀-E)]/ℏ; with E=12 eV, V₀=25 eV, L=0.05 nm, compute κ and transmission coefficient ≈ 10⁻⁴ to 10⁻⁵
• Part (d): Calculate Larmor frequency ω_L = eB/2mₑ = 4.40×10¹⁰ rad/s (or ν_L = 7.0×10⁹ Hz); then Zeeman splitting Δṽ = eB/4πmₑc = 0.467 cm⁻¹ (normal triplet separation)
• Part (e): Use rigid rotor formula E_J = BJ(J+1) where first line is 2B = 21.18 cm⁻¹; find B = 10.59 cm⁻¹, then I = ℏ/4πcB; calculate reduced mass μ = m_H m_Cl/(m_H+m_Cl); finally r₀ = √(I/μ) ≈ 1.27 Å

Begin with explicit statement of Pauli spin matrices, then systematically prove all three identities in part (a) showing anti-commutation relations, ladder operator properties, and cyclic permutation; for part (b) apply Heisenberg uncertainty principle with proper unit conversions from cm to meters; for part (c) set up the infinite square well wavefunction, identify n=1 ground state, and integrate probability density over the specified interval at box center. Allocate approximately 40% time to part (a) [20 marks], 30% to part (b) [15 marks], and 30% to part (c) [15 marks], ensuring all numerical answers carry proper units and significant figures.

• Explicit definition of Pauli matrices σₓ, σᵧ, σᵤ with standard matrix forms and their anti-commutation relations {σᵢ,σⱼ}=2δᵢⱼ
• Proof of (a)(i): σₓσᵧ+σᵧσₓ=0 etc. by direct matrix multiplication showing off-diagonal cancellation
• Proof of (a)(ii): σ₊σ₋=2(1+σᵤ) using σ₊=σₓ+iσᵧ, σ₋=σₓ−iσᵧ and σₓ²=σᵧ²=I
• Proof of (a)(iii): σₐσᵦ=iσᵧ (cyclic) using commutation [σᵢ,σⱼ]=2iεᵢⱼₖσₖ and anti-commutation results
• Part (b): Δp≥ℏ/(2Δx) with Δx=0.02 cm=2×10⁻⁴ m; electron and α-particle velocity uncertainties using mₑ=9.11×10⁻³¹ kg, mₐ=6.64×10⁻²⁷ kg
• Part (c): ψₙ(x)=√(2/L)sin(nπx/L) for 0<x<L, L=50Å=5×10⁻⁹ m; probability P=∫|ψ₁|²dx from 17.5Å to 32.5Å using sin² integral identity

This is a calculation-heavy question demanding precise numerical work for (a) and (c), plus conceptual explanation for (b). Allocate approximately 40% time to part (a) for Raman shift calculations and wavelength conversions, 30% to part (b) for explaining spin-orbit coupling mechanism and H-atom fine structure, and 30% to part (c) for L-S and j-j coupling term calculations. Begin with clear statements of formulas, show all intermediate steps, and conclude with physical significance of results.

• Part (a): Correct application of Raman shift formula Δν̃ = (1/λ₀ - 1/λs) in m⁻¹; calculation of Stokes and anti-Stokes wavelengths for new exciting line using the same Raman shift
• Part (b): Explanation of spin-orbit coupling as interaction between electron's spin magnetic moment and orbital magnetic field; derivation of fine structure splitting in H-atom showing ΔE ∝ j(j+1) - l(l+1) - s(s+1)
• Part (c)(i): L-S coupling: L = |l₁ - l₂| to l₁ + l₂ = 2,3,4,5,6; S = 0,1; J values from |L-S| to L+S for each combination
• Part (c)(ii): j-j coupling: j₁ = l₁ ± ½ = 7/2, 9/2; j₂ = l₂ ± ½ = 3/2, 5/2; J from |j₁-j₂| to j₁+j₂
• Physical interpretation: Term symbols notation ²ˢ⁺¹L_J for L-S coupling; comparison of coupling schemes showing L-S dominates for light atoms, j-j for heavy atoms

Begin with the relativistic energy-momentum relation to derive the de Broglie wavelength formula in part (a), then transition to quantum mechanical probability calculations for the harmonic oscillator in part (b), and conclude with a systematic exposition of Zeeman effects and degeneracy lifting in part (c). Allocate approximately 30% effort to part (a) due to its dual derivation-cum-calculation demand, 30% to part (b) for its integration-heavy probability analysis, and 40% to part (c) given its higher mark weightage and conceptual depth requiring clear energy level diagrams.

• Part (a): Derivation using E² = p²c² + m₀²c⁴ with E = K + m₀c², leading to λ = hc/√[K(K+2m₀c²)]; calculation of electron wavelength at 2 MeV (≈ 0.0056 Å or similar); non-relativistic limit reduction to λ = h/√(2m₀K)
• Part (b): Classical turning points for ground state SHO at x = ±√(ℏ/mω); probability integral P = ∫|ψ₀|²dx between -x₀ to +x₀ using Gaussian integral; demonstration that P ≈ 0.84 inside, hence ~16% outside classical limits
• Part (c): Normal Zeeman effect (spin singlet, orbital only, triplet splitting) vs anomalous Zeeman effect (spin-orbit coupling, Paschen-Back transition); role of g-factor (Landé g-factor for anomalous case); lifting of l-degeneracy in hydrogen via m_l quantum number and selection rules Δm = 0, ±1
• Clear distinction between strong and weak field regimes for Zeeman effects, with mention of Indian physicist S. Panck or relevant experimental context where applicable
• Proper handling of units throughout (MeV, Å, natural constants) and dimensional consistency checks

Solve each sub-part systematically with clear physical reasoning. For (a), apply Bragg's law correctly with order considerations; for (b), derive the specific heat expression rigorously and apply low-T approximation; for (c), explain decay mechanisms using strong vs. weak interaction; for (d), identify hadron properties from quark content; for (e), define chain reaction concepts with criticality condition. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e).

• For (a): Correct application of Bragg's law 2d sinθ = nλ to both cases, equating interplanar spacing d to find λ₁ = 0.168 nm or 1.68 Å
• For (b): Proper differentiation of U with respect to T to obtain C, and valid exponential approximation e^(ℏω/k_BT) >> 1 at low T showing C ∝ e^(-ℏω/k_BT)
• For (c): Explanation that ρ⁰ decays via strong interaction (resonance, ~10⁻²³s) while K⁰ decays via weak interaction (strangeness violation, ~10⁻¹⁰s), both to π⁺π⁻ final state
• For (d): Identification of (a) π⁺ (uđ), (b) π⁻ (ūd), (c) Ξ⁰ or neutron-like (dds = -1 charge, strangeness -1), (d) Ξ⁻ (uss: charge -1, strangeness -2, baryon number 1)
• For (e): Definition of chain reaction as self-sustaining neutron-induced fission sequence; critical size as minimum dimension where neutron multiplication factor k=1 (production = loss)

Begin with a brief introduction on nuclear structure studies in Indian context (Bhabha's contributions). For part (a), derive the Schrödinger solution using square well potential, showing binding energy ~2.2 MeV proves loose binding; allocate ~40% time. For (b), derive the oscillator spacing using nuclear radius relation R=r₀A^(1/3) and equating Fermi energy to ℏω; allocate ~30% time. For (c), enumerate three neutrino flavors with mass hierarchy and mention Indian experiments like INO; allocate ~30% time. Conclude with significance for nuclear astrophysics.

• Part (a): Importance of deuteron as only two-nucleon bound system, simplest test of nuclear forces; solution of radial Schrödinger equation for l=0 with finite square well; boundary conditions at r=R; obtain transcendental equation; show binding energy E_b≈2.224 MeV << typical nuclear energy scale (~8 MeV/nucleon) proving loose binding
• Part (a): Mention deuteron's large size (r_d≈4.3 fm), no excited bound states, electric quadrupole moment indicating non-spherical shape and tensor force importance
• Part (b): Derivation starting from 3D isotropic harmonic oscillator with Hamiltonian H=p²/2m + ½mω²r²; energy levels E_N=(N+3/2)ℏω where N=2(n-1)+l; major shells at N=0,1,2,3...
• Part (b): Relate ℏω to nuclear size using R=r₀A^(1/3) and Fermi momentum; equate ℏ²k_F²/2m to ℏω scaling; derive ℏω≈41A^(-1/3) MeV with r₀≈1.2 fm
• Part (c): Three types—electron neutrino (ν_e), muon neutrino (ν_μ), tau neutrino (ν_τ); mass differences from neutrino oscillation experiments; solar neutrino puzzle and KamLAND, Super-Kamiokande evidence
• Part (c): Mass hierarchy: normal (m₁<m₂<m₃) vs inverted; Δm²₂₁≈7.5×10⁻⁵ eV², |Δm²₃₁|≈2.5×10⁻³ eV²; mention upper limits (~eV scale) and cosmological constraints; Indian Neutrino Observatory relevance

Calculate requires systematic numerical problem-solving across all three sub-parts. Allocate approximately 40% of effort to part (a) as it carries 20 marks and involves DC biasing analysis with feedback; 35% to part (b) for quantum mechanical angular momentum calculations; and 25% to part (c) for the FET pinch-off voltage derivation. Begin each part with the relevant formula, show complete derivation with substituted values, and conclude with proper units and physical significance.

• Part (a): Correct application of KVL to base-collector feedback loop with Ic = βIb and proper handling of the 100 kΩ feedback resistor to establish quiescent operating point
• Part (a): Recognition that VBE = 0 simplifies analysis to finding IB from Vcc = IB·RB + IE·RL relationship with proper current relationships
• Part (b): Correct determination of L, S, J values for 3d¹ configuration of Scandium (L=2, S=½, J=3/2, 5/2) and selection of lowest energy state using Hund's third rule for less than half-filled shell
• Part (b): Accurate calculation of Landé g-factor using g = 1 + [J(J+1) + S(S+1) - L(L+1)]/[2J(J+1)] and subsequent Zeeman energy splitting ΔE = g·μB·B·MJ
• Part (c): Proper application of pinch-off voltage formula VP = (q·ND·a²)/(2ε) with correct unit conversion from cm to m and use of relative permittivity ε = εr·ε0
• Part (c): Recognition that channel width 2a = 6×10⁻⁴ cm gives a = 3×10⁻⁶ m for the depletion region calculation

Begin with the directive 'calculate' for part (b) which carries the highest marks (20), then address 'sketch' for part (a) (10 marks) and 'explain/discuss' for part (c) (20 marks). Allocate approximately 35% time to part (b) for rigorous derivation of magnetic susceptibility using Hund's rules and Curie law, 25% to part (a) for accurate load line construction with proper intercepts, and 40% to part (c) for comprehensive explanation of internal conversion with coefficient definition and conditions. Structure as: (a) circuit analysis → load line sketch; (b) quantum mechanical derivation → numerical substitution; (c) phenomenon explanation → coefficient definition → conditions discussion.

• Part (a): Correct identification of DC load line endpoints (cut-off and saturation points) from circuit parameters, proper axes labeling (V_CE vs I_C), and accurate linear plot showing Q-point placement
• Part (b): Application of Hund's rules to determine total angular momentum J for Nd³⁺ (4f³ configuration), calculation of Landé g-factor, derivation of effective magnetic moment, and correct substitution into Curie law formula χ = μ₀Nμ_eff²/(3k_BT) with proper unit handling
• Part (b): Correct numerical evaluation yielding χ ≈ 4.5 × 10⁻³ (dimensionless SI) or equivalent, showing all intermediate steps for μ_eff, μ_B conversion, and temperature dependence
• Part (c): Clear explanation of internal conversion as radiationless nuclear de-excitation with energy transfer to orbital electron, proper definition of α_IC = N_e/N_γ (conversion electrons to gamma photons ratio)
• Part (c): Discussion of conditions favoring internal conversion: high Z nuclei, low transition energy, large change in nuclear angular momentum (ΔL ≥ 2), and Mössbauer spectroscopy relevance for Indian nuclear physics research