UPSC Physics 2022 — Paper II

Solve requires systematic problem-solving across all five sub-parts with equal time allocation (~20% each). Begin with concise conceptual definitions for (a), then proceed to step-by-step calculations for (a)-(c), rigorous derivations for (d)-(e). Structure: direct answers without elaborate introduction, showing all intermediate steps, unit conversions, and final boxed results for each sub-part.

• (a) State de Broglie hypothesis λ = h/p; derive λ = h/√(2mE) for non-relativistic case; calculate He wavelength using m_He = 4m_p (alpha particle mass) and E = 300 eV, obtaining λ ≈ 8.3×10⁻¹² m
• (b) Apply infinite square well energy levels E_n = n²π²ℏ²/(8ma²) for well width 2a; use ΔE = E₄ - E₂ = hf to solve for a, obtaining a ≈ 1.5×10⁻⁹ m or width 2a ≈ 3 nm
• (c) Apply NMR resonance condition hν = gₙμₙB where μₙ = eℏ/(2m_p); solve for B = hν/(gₙμₙ) ≈ 2.82 T
• (d) Derive g_L = 1 from μ_L = -(e/2m)L and g_S = 2 from μ_S = -(e/m)S; apply Landé formula g_J = 1 + [J(J+1)+S(S+1)-L(L+1)]/[2J(J+1)] for ³P₁ (L=1,S=1,J=1) to get g_J = 3/2
• (e) Prove [J_z, J_+] = ℏJ_+ and [J_z, J_-] = -ℏJ_- using [J_z, J_x] = iℏJ_y and [J_z, J_y] = -iℏJ_x; prove J_-J_+ = J² - J_z² - ℏJ_z by expanding and using J_x² + J_y² = J² - J_z²

Derive the tunneling probability for a 1D potential barrier in part (a) using boundary condition matching; prove the virial theorem relation for hydrogen in (b); enumerate n² degeneracy with proper quantum number counting for (c)(i); and decode the term symbol ^4D_{5/2} to extract L, S, J and suggest a minimal electron configuration for (c)(ii). Allocate approximately 40% time to (a), 30% to (b), and 30% combined to both parts of (c), ensuring each sub-part receives proportional attention to its marks.

• Part (a): Set up time-independent Schrödinger equation for regions I (x<0), II (0<x<a), and III (x>a) with appropriate wave functions; apply continuity conditions for ψ and dψ/dx at x=0 and x=a; obtain transmission coefficient T = |F/A|² = [1 + (k₁²+k₂²)²sinh²(κa)/(4k₁²k₂²)]⁻¹ ≈ 16E(V₀-E)/V₀² exp(-2κa) for E<V₀
• Part (b): Apply virial theorem 2⟨T⟩ = ⟨r·∇V⟩ for Coulomb potential V∝1/r; show ⟨T⟩ = -Eₙ/2 and ⟨V⟩ = 2Eₙ; conclude Eₙ = ⟨V⟩/2 + ⟨T⟩ = ⟨V⟩ - ⟨V⟩/2 = ⟨V⟩/2 + ⟨V⟩/2 verification or direct proof using hydrogenic wave functions and expectation value ⟨r⁻¹⟩ = 1/(n²a₀)
• Part (c)(i): Count states using quantum numbers n, l, m_l: for given n, l ranges 0 to n-1; each l has (2l+1) m_l values; sum Σ(2l+1) from l=0 to n-1 equals n²; alternatively include spin degeneracy factor of 2 to get 2n² for total states
• Part (c)(ii): Decode term symbol ^4D_{5/2}: 2S+1=4 gives S=3/2; D means L=2; J=5/2; minimum electrons = 3 (for S=3/2 requires at least 3 unpaired electrons); possible configuration: [Ar]3d³4s² (vanadium) or [Ar]3d⁷4s² (cobalt) or 2p²3p¹ excited state
• Physical significance: Connect tunneling to α-decay (Gamow factor), STM microscopy, and quantum devices; relate hydrogen result to stability of atomic orbits; connect term symbols to atomic spectroscopy and Hund's rules applications

This question demands clear explanation across theoretical derivation, experimental physics, and spectroscopic principles. Allocate approximately 30% time to part (a) for rigorous spinor derivation using rotation matrices, 40% to part (b) covering Stern-Gerlach significance, schematic diagram with shaped pole pieces, and silver atom rationale, and 30% to part (c) for Franck-Condon principle with potential energy curve diagrams. Begin with mathematical formulation, transition to experimental verification, and conclude with spectroscopic application.

• Part (a): Derivation of spin wave function χ₊(η) = cos(θ/2)χ₊ + e^(iφ)sin(θ/2)χ₋ using direction cosines (θ,φ) of unit vector η, showing eigenvalue equation S·η χ = +(ℏ/2)χ
• Part (b)(i): Explanation of Stern-Gerlach importance—first direct evidence of space quantization, electron spin discovery, basis for quantum measurement theory and quantum information (qubit realization)
• Part (b)(ii): Schematic diagram showing oven, collimating slits, inhomogeneous magnetic field with pole pieces shaped as wedge/cylinder or knife-edge to create ∂Bz/∂z ≠ 0, and detection screen
• Part (b)(iii): Silver atom rationale—single valence electron (5s¹) with J=1/2, zero orbital angular momentum (L=0), no hyperfine splitting complication, high atomic mass minimizing Doppler broadening, thermal vaporization feasibility
• Part (c): Franck-Condon principle statement (vertical transitions with nuclear positions fixed), explanation using overlap integrals of vibrational wavefunctions |⟨ψv'|ψv''⟩|², intensity distribution in progression and sequences with Condon parabola

Begin with the directive 'calculate' for the numerical sub-parts (a)(ii), (b)(ii), and (c), while using 'explain', 'draw', and 'obtain' for the descriptive sub-parts. Allocate approximately 40% of time/effort to part (a) given its 20 marks, 30% each to parts (b) and (c). Structure as: brief conceptual introduction → systematic treatment of each sub-part with clear labeling → concluding synthesis on how isotope effects and quantum mechanical treatments reveal molecular structure.

• For (a)(i): Explain that isotopic substitution increases reduced mass μ, decreases rotational constant B (B ∝ 1/μ), and compresses rotational energy levels leading to closer-spaced spectral lines
• For (a)(ii): Calculate B_H₂ = ℏ/(4πcI_H₂) and B_D₂ = ℏ/(4πcI_D₂), showing I_D₂ = 2I_H₂ due to doubled reduced mass, hence B_D₂ = B_H₂/2, or explicitly compute numerical values
• For (a)(iii): Draw energy level diagrams showing equally spaced levels for rigid rotor vs. diverging levels for non-rigid rotor (centrifugal distortion), with corresponding spectral line spacing
• For (b)(i): Explain anharmonicity introduces cubic term in potential, causes energy level convergence, overtone frequencies as ν₀(1-2x_e), (ν₀-4ν₀x_e), etc., and decreased spacing at higher v
• For (b)(ii): Calculate rotational period T = 2π/ω = 2πI/√[J(J+1)]ℏ using given I = 0.0264×10⁻⁴⁵ kg.m² and J=3, obtaining T ≈ 1.3×10⁻¹³ s or equivalent
• For (c): Derive normalized eigenvectors of σₓ = (0 1; 1 0) as (1/√2)(1; ±1) and σᵧ = (0 -i; i 0) as (1/√2)(1; ±i), showing explicit normalization and orthogonality

The directive 'explain' demands clear causal reasoning and mechanistic understanding across all sub-parts. Allocate approximately 20% time each to (a), (b), (c), (d), and (e) as all carry equal marks. Structure with brief introductions for each sub-part, followed by systematic causal explanations, and conclude with synthesizing remarks on nuclear forces, quark confinement, and device applications. For (b)(i)-(ii), treat both as mandatory; for (e), ensure all three components (NAND/NOR universality, XOR diagram, Boolean equation, truth table) are addressed.

• (a) Explains why dineutron is unbound despite charge-independent nuclear force: Pauli exclusion principle forbids identical fermions in same state, so nn system must be in spin-triplet (antisymmetric space) with reduced overlap vs deuteron's spin-singlet; cites tensor force and spin-dependence of nuclear force
• (b)(i) Demonstrates baryon spin-1 impossibility: three quarks (spin-1/2 each) can only combine to total spin 1/2 or 3/2 via angular momentum coupling, never spin-1; shows explicit quark spin wavefunction construction
• (b)(ii) Proves anti-baryon charge +2 impossibility: anti-baryon has three antiquarks, each with charge +1/3 or +2/3, giving maximum possible charge +2 only if all three are anti-up (+2/3 each), but this violates color singlet requirement (antisymmetric color × antisymmetric flavor × symmetric spin)
• (c) Contrasts Type-I (complete Meissner effect, H < Hc) vs Type-II (vortex state, Hc1 < H < Hc2): explains flux penetration via Abrikosov vortices, higher critical fields enabling superconducting magnets; mentions Indian contributions (K. S. Novoselov's graphene work context, or indigenous superconductor research at BARC/IITs)
• (d) Defines FET as unipolar (single carrier type: electrons in n-channel or holes in p-channel) vs BJT (bipolar: both carriers); lists superiority: high input impedance, voltage-controlled, negligible gate current, thermal stability, smaller size, faster switching; mentions Indian semiconductor initiatives (ISRO's indigenous fabrication)
• (e) Proves NAND/NOR universality by constructing NOT, AND, OR from each; provides complete XOR: logic diagram with NAND gates, Boolean expression Y = A⊕B = A'B + AB' = (A+B)(A'+B'), truth table with all four input combinations

Begin with a concise introduction linking nuclear structure, crystallography and nuclear stability. For part (a), derive the degeneracy formula g_N = (N+1)(N+2)/2 and identify magic numbers 2, 8, 20, 28, 50, 82, 126 using N = 2(n-1)+l. For part (b), calculate packing fractions π/6, π√3/8 and π√2/6 with clear unit cell diagrams. For part (c), state the full Weizsäcker formula with all seven terms, then apply Q-value calculations to show ²³⁸U prefers α-decay over β⁻ due to higher binding energy per nucleon and Coulomb barrier considerations. Allocate ~35% effort to (a), ~30% to (b), and ~35% to (c) including the comparative analysis.

• Part (a): Derivation of total quantum number N = 2(n-1) + l and proof that degeneracy g_N = (N+1)(N+2)/2 for 3D isotropic harmonic oscillator
• Part (a): Enumeration of shell closures at N = 0,1,2,3,4,5,6 leading to magic numbers 2, 8, 20, 40, 70, 112, 168; correction to observed magic numbers 2, 8, 20, 28, 50, 82, 126 with spin-orbit coupling mention
• Part (b): Calculation of atoms per unit cell (1, 2, 4) and packing fractions (π/6 ≈ 0.52, π√3/8 ≈ 0.68, π√2/6 ≈ 0.74) for SC, BCC, FCC with proper geometric derivation involving atomic radius and lattice parameter relations
• Part (c): Complete Weizsäcker formula with volume, surface, Coulomb, asymmetry, pairing, shell correction and pairing terms with correct coefficients; physical explanation of each term
• Part (c): Q-value calculation for α-decay (Q_α ≈ 4.27 MeV, spontaneous) versus β⁻-decay energetics; explanation using N/Z ratio, Coulomb repulsion dominance, and liquid drop model predictions for heavy nuclei (A>210)

Begin with a brief introduction acknowledging the historical development of quantum theories of specific heat. For part (a), spend approximately 40% of effort explaining Einstein's assumption of independent oscillators and Debye's improvement using a continuous spectrum of phonon modes with the Debye cutoff. For part (b), allocate 30% to derive the photon energy using mass-energy equivalence and binding energy of deuteron (~2.224 MeV), then assess recoil energy significance. For part (c), use remaining 30% to contrast resistance-temperature curves and explain Cooper pair formation via electron-phonon interaction. Conclude by noting the unifying theme of quantum effects in condensed matter.

• Part (a): Einstein's theory assumes all atoms vibrate with same frequency ν_E, leading to C_V → 0 exponentially at low T instead of T³ law; Debye treats solid as continuous elastic medium with cutoff frequency ν_D, introducing Debye temperature Θ_D and density of states g(ν) ∝ ν²
• Part (a): Debye's integral expression for specific heat and its correct prediction of C_V ∝ T³ at T << Θ_D and Dulong-Petit law at T >> Θ_D; mention limitations at intermediate T
• Part (b): Application of mass-energy conservation: E_γ = [m_n + m_p - m_d]c² = B_d (binding energy of deuteron) ≈ 2.224 MeV; recoil energy E_recoil = E_γ²/(2m_dc²) ≈ 1.3 keV is negligible (~0.06%)
• Part (c): Normal conductor: resistance increases with T due to enhanced phonon scattering (ρ ∝ T for T > Θ_D, ρ ∝ T⁵ at low T); superconductor: zero resistance below T_c with sharp transition
• Part (c): Cooper pair formation via attractive interaction mediated by phonon exchange; electrons with opposite k and spin form bound state with energy gap; coherence length and BCS theory essence

Derive the Fermi level position for n-type semiconductor in part (a) using charge neutrality and appropriate approximations at low temperature. For part (b), apply Bragg's law to find interplanar spacing, then use fcc geometry and density formula to calculate Avogadro number. For part (c), analyze each decay using conservation laws (charge, baryon number, lepton number, strangeness, energy-momentum) to determine allowed/forbidden status and identify the responsible interaction. Allocate approximately 40% time to (a), 30% to (b), and 30% to (c) based on mark distribution.

• Part (a): Derivation showing E_F = (E_C + E_D)/2 using charge neutrality condition n_0 = n_D+ at low T, with proper assumptions about donor ionization and negligible intrinsic carriers
• Part (b): Application of Bragg's law (2d sin θ = nλ), calculation of d_111 for fcc, relation between lattice parameter and atomic density, and final calculation of N_A ≈ 6.02 × 10^23 mol^-1
• Part (c)(i): n → p + e⁻ + ν̄ₑ is allowed weak decay (β-decay) conserving all quantum numbers
• Part (c)(ii): Λ° → π⁺ + π⁻ is forbidden as it violates baryon number conservation (B=1 → B=0)
• Part (c)(iii): π⁻ → e⁻ + γ is forbidden as it violates lepton number conservation (L=0 → L=1) and angular momentum
• Part (c)(iv): π° → e⁻ + e⁺ + νₑ + ν̄ₑ is forbidden as it violates energy conservation (m_π° < 2m_e + E_ν)
• Part (c)(v): π⁺ → e⁻ + e⁺ + μ⁺ + νᵤ is forbidden as it violates charge conservation (+1 → +1-1+1+0 = +1 actually check: +1 → -1+1+1+0 = +1, but violates lepton family number and energy)