UPSC Physics 2023 — Paper II

Calculate requires systematic numerical working with proper physical reasoning. Structure: (a) Apply E₁ = π²ℏ²/(2mL²) for all three cases, showing orders of magnitude comparison; (b) Complete the square for the shifted harmonic oscillator, finding new equilibrium and energy levels; (c) Solve radial Schrödinger equation for l=0 with boundary conditions at r=a; (d) Use force method or quantum expectation for Stern-Gerlach splitting; (e) Apply Larmor precession formula τ = μ×B. Allocate ~20% time each to (a), (b), (c), (d), (e) as all carry equal marks.

• (a) Zero-point energy calculation: E₁ = π²ℏ²/(2mL²) for 100g ball (~10⁻⁶⁷ J), oxygen atom (~10⁻²¹ J), electron (~10⁻¹⁸ J); explicit comparison showing macroscopic E₁ is negligible vs thermal energy kT ~ 10⁻²¹ J at 300K
• (b) Shifted harmonic oscillator: complete square to get H = p²/2m + ½mω²(X - qE/mω²)² - q²E²/2mω²; energy levels Eₙ = (n+½)ℏω - q²E²/2mω²; wave functions ψₙ(x) = φₙ(x - qE/mω²) where φₙ are standard HO eigenfunctions
• (c) Spherical well: radial equation for l=0 with u(r)=rR(r); boundary conditions u(0)=0, continuity of u and u' at r=a; transcendental equation for bound states; condition for two bound states requires at least one node, giving minimum depth V₀ ≥ π²ℏ²/(8ma²)
• (d) Stern-Gerlach: force F = μ_z(dB/dz) = ±(eℏ/2m_e)(dB/dz); classical trajectory with transverse acceleration; deflection δz = (F/m_H)(L/v)²/2 where v = √(3kT/m_H); numerical value ~0.3-0.5 mm
• (e) Larmor precession: ω_L = g_lμ_BB/ℏ = μ_BB/ℏ for orbital moment (g_l=1); torque |τ| = μBsinθ = lμ_B·B·sin30°; precession rate ~10⁹ rad/s, torque ~10⁻²³ Nm

Solve this multi-part quantum mechanics problem by first proving the simultaneous measurability in part (a) through operator commutation, then calculating reflection coefficient in part (b) using boundary conditions, and finally computing energy levels and velocities for both quantum and classical systems in part (c). Allocate approximately 30% time to part (a), 25% to part (b), and 45% to part (c) given its higher mark weightage and computational complexity. Conclude with a comparative discussion highlighting why quantum effects dominate at atomic scales but are negligible for macroscopic objects.

• Part (a): Prove [P, S²] = 0 and [P, Sz] = 0 using σ⃗₁·σ⃗₂ = (S²/ℏ²) - 3/2 and [S², Sz] = 0, establishing complete set of commuting observables
• Part (b): Apply boundary conditions at x=0 for wavefunctions ψ_I = Ae^(ikx) + Be^(-ikx) and ψ_II = Ce^(ik'x) with k=√(2mE)/ℏ, k'=√(2m(E-V))/ℏ=√(2mE/4)/ℏ to find reflection coefficient R = |B/A|² = (k-k')²/(k+k')² = 1/9
• Part (c)(i): Calculate E_n = n²π²ℏ²/(2ma²) for electron (m=9.11×10⁻³¹ kg, a=10⁻¹⁰ m): E₁≈6.02×10⁻¹⁸ J≈37.6 eV, E₂≈150.4 eV, E₃≈338.4 eV
• Part (c)(ii): Calculate same formula for metallic sphere (m=10⁻³ kg, a=0.1 m): E₁≈5.49×10⁻⁶⁴ J, showing extremely closely spaced levels
• Part (c)(b): Discuss quantum effects: electron shows discrete levels, zero-point energy, wave-particle duality; sphere behaves classically with continuous energy spectrum
• Part (c)(c): Apply ΔxΔp≥ℏ/2 to estimate v_electron≈1.1×10⁶ m/s (relativistic correction may be noted) and v_sphere≈5×10⁻³¹ m/s (effectively at rest)
• Demonstrate understanding that quantum-classical correspondence principle requires n→∞ or ℏ→0 limits

Begin with a brief introduction linking atomic structure to spectroscopic phenomena. For part (a), spend ~30% time explaining vector atom model concepts and Stern-Gerlach experimental validation. For part (b), allocate ~30% to deriving Landé g-factor with proper LS coupling for ³P₁ state and calculating Zeeman splitting. For part (c), dedicate ~40% to quantum theory of Raman effect with rotational energy level diagrams. Conclude by emphasizing how these three phenomena collectively demonstrate space quantization and quantum mechanical treatment of atomic systems.

• Part (a): Vector atom model with orbital (L), spin (S) and total (J) angular momentum vectors; Stern-Gerlach experiment showing space quantization and electron spin discovery through Ag beam splitting
• Part (a): Explanation of how Stern-Gerlach results confirmed quantization of magnetic moment and existence of spin angular momentum, resolving Bohr-Sommerfeld model limitations
• Part (b): Landé g-factor formula derivation for LS coupling: g_J = 1 + [J(J+1) + S(S+1) - L(L+1)]/[2J(J+1)]; calculation for ³P₁ (L=1, S=1, J=1) yielding g_J = 3/2
• Part (b): Zeeman energy shift ΔE = μ_B·g_J·B·m_J with m_J = -1,0,+1; numerical evaluation for B=0.1 T giving splitting of ±9.27×10⁻²⁴ J or ±5.79×10⁻⁵ eV
• Part (c): Raman effect as inelastic light scattering with Stokes/anti-Stokes lines; quantum theory involving virtual energy levels and polarizability tensor
• Part (c): Rotational Raman spectrum showing ΔJ = ±2 selection rule, spacing of 4B, and alternating intensity due to nuclear spin statistics (relevant for homonuclear molecules like N₂, O₂)
• Part (c): Distinction between Rayleigh, Stokes and anti-Stokes lines with energy level diagram showing rotational transitions

Solve this multi-part numerical-cum-descriptive question by allocating approximately 30% time to part (a) on particle-in-a-box normalization and momentum expectation, 30% to part (b) on hydrogen 2p state radial probability, and 40% to part (c) on NMR principles and MRI applications. Begin each part with the relevant formula, show complete derivation steps, and conclude with physical interpretation—especially connecting MRI to healthcare applications in Indian context like AIIMS Delhi's advanced imaging facilities.

• Part (a): Normalization constant A = √(2/L) obtained by integrating |ψ|² from 0 to L; momentum expectation value ⟨p⟩ = 0 shown via direct integration or operator method
• Part (a): Recognition that ⟨p⟩ = 0 reflects stationary state with equal probability of left/right motion, or explicit calculation using p̂ = -iℏ(d/dx)
• Part (b): Radial wave function R₂₁(r) ∝ r·exp(-r/2a₀) for 2p state; radial probability density P(r) = r²|R₂₁|² ∝ r⁴exp(-r/a₀)
• Part (b): Most probable distance r_mp = 4a₀ obtained by dP/dr = 0; maximum probability density value P(r_mp) = (1/24a₀)·(4/e)⁴ or equivalent simplified form
• Part (c): NMR defined as resonant absorption of RF radiation by nuclear spins in magnetic field; working principle involving Zeeman splitting, Larmor precession, and resonance condition ω = γB₀
• Part (c): MRI working: gradient coils for spatial encoding, RF pulses for excitation, detection of FID signals; T₁/T₂ contrast for tissue differentiation; Indian relevance: indigenous MRI development at BARC, widespread diagnostic use for cancer and neurological disorders

Begin with a brief introduction acknowledging the breadth from particle physics to solid state physics. For part (a), explain helicity, lepton number conservation, and scattering experiments; for (b), set up the radioactive decay equation and solve for age; for (c), classify all ten reactions by interaction type using conservation laws; for (d), derive Laue conditions from reciprocal lattice vectors; for (e), use charge neutrality and mass action law to show Fermi level shifts. Allocate time proportionally: ~15% each for (a), (b), (c), and 20% each for (d) and (e) due to derivations required.

• (a) Distinguishing νₑ and ν̄ₑ: helicity differences, lepton number conservation (Lₑ = +1 vs −1), and experimental evidence from inverse beta decay (ν̄ₑ + p → n + e⁺) vs (νₑ + n → p + e⁻)
• (b) Age calculation: determine initial ¹⁴C mass from given ratio, apply N = N₀e^(-λt) with λ = ln(2)/T₁/₂, relate activity A = λN, solve for t ≈ 11,400 years
• (c) Interaction classification: (i) Weak, (ii) Strong, (iii) Strong (resonance formation), (iv) Electromagnetic, (v) Weak, (vi) Weak (strangeness changing), (vii) Electromagnetic, (viii) Weak, (ix) Weak, (x) Electromagnetic
• (d) Reciprocal lattice derivation: define reciprocal basis vectors b₁, b₂, b₃, show scattering vector Δk = G (reciprocal lattice vector), derive Laue conditions k·G = G²/2 or equivalently 2k·G = G²
• (e) Fermi level shifts: for n-type, n = N_D⁺ + nᵢ²/n, show E_F moves toward E_C; for p-type, p = N_A⁻ + nᵢ²/p, show E_F moves toward E_V using charge neutrality and mass action law

Derive the Rutherford differential cross-section formula in part (a) by setting up the scattering geometry, applying Coulomb's law, and integrating to obtain the solid angle dependence. For part (b), apply the particle-in-a-box quantum mechanical model with appropriate boundary conditions to calculate the Fermi energy for Fe-56 nucleons. In part (c), define nuclear forces with their key characteristics, then explain Yukawa's meson exchange theory with mass-energy relation and range estimation. Allocate approximately 40% effort to (a), 30% to (b), and 30% to (c) based on mark distribution.

• Part (a): Coulomb scattering geometry with impact parameter b, scattering angle θ, and hyperbolic trajectory; derivation of relation b = (kZe²/2E)cot(θ/2); differential cross-section dσ/dΩ = (kZe²/4E)²cosec⁴(θ/2); assumptions: point charge nucleus, single scattering, non-relativistic α-particles, neglect of electron screening
• Part (a): Integration over solid angle to show total cross-section divergence, physical significance of Rutherford formula validation through Geiger-Marsden experiments at Manchester
• Part (b): Particle in a 3D cubic box energy levels E = (h²/8mL²)(nx²+ny²+nz²); nuclear diameter L = 10⁻¹² cm; neutron and proton as spin-½ fermions with degeneracy g=2; Fermi momentum and energy calculation for A=56, Z=26, N=30
• Part (b): Numerical computation with ħc = 197 MeV-fm, nucleon mass ≈ 938 MeV/c², proper unit conversion from cm to fm; comparison with empirical nuclear Fermi energy ~38 MeV
• Part (c): Nuclear force characteristics: short-range (~1-3 fm), spin-dependent, charge-independent, saturated, non-central tensor component; contrast with Coulomb force
• Part (c): Yukawa meson theory: virtual particle exchange, uncertainty principle ΔE·Δt ≈ ħ, meson mass mπ from range R ≈ ħ/mπc ≈ 1.4 fm giving mπ ≈ 140 MeV/c²; prediction of π-meson later discovered in cosmic rays by Powell (1950 Nobel Prize)

The directive 'explain' demands clear exposition with logical flow across all three parts. Allocate approximately 30% time/words to part (a) on diamagnetism (15 marks), 40% to part (b) on Debye theory (20 marks), and 30% to part (c) on Wien-Bridge oscillator (15 marks). Structure: begin each part with defining the core concept, proceed through derivations with intermediate steps shown, and conclude with physical significance and limitations.

• Part (a): Larmor precession explanation, derivation of χ = -μ₀NZe²⟨r²⟩/(6mₑ), proportionality to Z via electron count, and explanation of why all electrons contribute (closed shells, no net paramagnetism)
• Part (b): Debye frequency distribution g(ω) ∝ ω², derivation of Cᵥ = 9R(T/θ_D)³∫₀^(θ_D/T) x⁴eˣ/(eˣ-1)²dx, T³ law at low T and Dulong-Petit at high T, comparison with Einstein's single frequency assumption
• Part (c): Wien-Bridge circuit with four resistors and two capacitors, frequency formula f = 1/(2πRC), Barkhausen criterion (R₃/R₄ = 2), amplitude stabilization via lamp or diodes
• Debye theory assumption: continuous spectrum of frequencies up to ω_D vs Einstein's single ω_E; Debye's acoustic phonon approximation
• Drawbacks of Debye theory: fails at intermediate temperatures, neglects optical phonons, assumes isotropic solids and linear dispersion
• Experimental verification: specific heat data for copper, lead, diamond showing T³ region and Debye temperature extraction

The directive 'explain' demands clear conceptual exposition with logical reasoning. Allocate approximately 40% effort to part (a) given its 20 marks, 30% to part (b) for 15 marks, and 30% combined to part (c)(i) and (ii) for 15 marks. Structure: begin with reactor physics fundamentals, transition to superconductivity phenomena, then conclude with systematic op-amp circuit analysis using negative feedback principles.

• Critical size definition: minimum dimensions for self-sustaining chain reaction where neutron production equals losses (multiplication factor k=1); mention critical mass and critical volume relationship
• Nuclear reactor main features: fuel (enriched U-235/Pu-239), moderator (heavy water/graphite in Indian PHWRs), control rods (Cd/B), coolant, shielding; reference Indian reactors like Dhruva or CIRUS
• Superconductivity: zero DC resistance below critical temperature Tc; Meissner effect as perfect diamagnetism with expulsion of magnetic field (B=0 inside); Type-I vs Type-II distinction
• Diamagnetic necessity: superconductors must expel magnetic flux to maintain zero resistance state; London penetration depth and thermodynamic argument for free energy minimization
• Op-amp input impedance with feedback: Z_in(CL) = Z_in(OL)[1+βA_OL] for non-inverting; output impedance reduction by factor (1+βA_OL)
• Closed-loop gain derivation: A_CL = A_OL/(1+βA_OL) ≈ 1/β for large A_OL; identify feedback network β from resistor configuration
• Numerical calculation: substitute given values Z_in=2MΩ, Z_out=75Ω, A_OL=200,000 with appropriate β determination from implied circuit topology