Physics 2025 Paper II 50 marks State

Q4

(a) State how for spin-half particles, the spin (σ) can be expressed by its three components σ_x, σ_y and σ_z. 20 marks (b) By applying the Schrödinger's equation to the ground state of hydrogen atom, determine the zero-point energy. 15 marks (c) Distinguish between fluorescence and phosphorescence. Explain the mechanisms responsible for these phenomena. Discuss the applications of fluorescence and phosphorescence in the fields such as biochemistry, material science, etc. 15 marks

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(a) बताइए कि किस प्रकार अर्ध-प्रचक्रण कणों के लिए प्रचक्रण (σ) को उसके तीन अवयवों σ_x, σ_y और σ_z द्वारा व्यक्त किया जा सकता है। 20 अंक (b) श्रोडिंगर समीकरण को हाइड्रोजन परमाणु की आध अवस्था के लिए प्रयुक्त करके शून्य-बिंदु ऊर्जा की गणना कीजिए। 15 अंक (c) प्रतिदीप्ति और स्कुरदीप्ति के बीच प्रभेद कीजिए। इन परिघटनाओं के लिए उत्तरदायी क्रियाविधियों की व्याख्या कीजिए। जैव रासायनिकी, पदार्थ विज्ञान इत्यादि जैसे क्षेत्रों में प्रतिदीप्ति और स्कुरदीप्ति के अनुप्रयोगों की चर्चा कीजिए। 15 अंक

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Directive word: State

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Approach

The directive 'state' in part (a) demands precise, formal presentation of spin-half algebra with Pauli matrices, while parts (b) and (c) require 'determine' and 'distinguish/explain/discuss' respectively. Allocate approximately 40% of effort to part (a) given its 20 marks: present σ = (σ_x, σ_y, σ_z) with explicit 2×2 Pauli matrices, their commutation relations, and eigenvalue properties. Spend ~30% on part (b): solve radial Schrödinger equation for n=1, l=0, derive E₁ = -13.6 eV and identify zero-point energy from kinetic term or uncertainty principle. Allocate remaining ~30% to part (c): construct clear comparison table, explain singlet vs. triplet states and intersystem crossing, cite Indian applications such as fluorescence microscopy at IISc Bengaluru or phosphorescent safety signage in Indian Railways. Conclude with integrated remarks on quantum phenomena spanning atomic to molecular scales.

Key points expected

  • Part (a): Pauli spin matrices σ_x = [[0,1],[1,0]], σ_y = [[0,-i],[i,0]], σ_z = [[1,0],[0,-1]]; spin operator S = (ℏ/2)σ; commutation relations [σ_i, σ_j] = 2iε_ijk σ_k; eigenvalues ±1 for each component
  • Part (a): Spin as vector operator σ = σ_x î + σ_y ĵ + σ_z k̂; total spin magnitude S² = s(s+1)ℏ² with s=1/2; connection to SU(2) representation
  • Part (b): Radial Schrödinger equation for hydrogen ground state: -ℏ²/2μ ∇²ψ - e²/4πε₀r ψ = Eψ; separation into radial and angular parts; ground state wavefunction ψ₁₀₀ = (1/√πa₀³) exp(-r/a₀)
  • Part (b): Zero-point energy derivation: E₁ = -μe⁴/8ε₀²ℏ² = -13.6 eV; kinetic energy expectation value ⟨T⟩ = +13.6 eV; potential energy ⟨V⟩ = -27.2 eV; zero-point energy identified as positive kinetic energy contribution or via ΔxΔp ≥ ℏ/2
  • Part (c): Distinction table: fluorescence (spin-allowed, S₁→S₀, 10⁻⁹-10⁻⁷ s) vs phosphorescence (spin-forbidden, T₁→S₀, 10⁻³-10³ s); Jablonski diagram with radiative and non-radiative transitions
  • Part (c): Mechanisms: fluorescence via prompt emission without change in spin multiplicity; phosphorescence requires intersystem crossing (ISC) with spin-orbit coupling, delayed emission through forbidden transition
  • Part (c): Applications: fluorescence—green fluorescent protein (GFP) tagging, flow cytometry, Indian biomedical research (CCMB Hyderabad); phosphorescence—organic light-emitting diodes (OLEDs), persistent luminescent materials for emergency exit signs, security inks in Indian currency

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5For (a): correctly identifies Pauli matrices as generators of SU(2), states eigenvalues ±ℏ/2 for S_z, and notes σ² = 3I. For (b): correctly identifies ground state energy -13.6 eV and distinguishes zero-point energy as kinetic energy contribution. For (c): accurately describes singlet and triplet states, ISC mechanism, and distinguishes timescales by 6+ orders of magnitude.States Pauli matrices without commutation relations; calculates ground state energy but confuses zero-point energy with total energy or ionization energy; describes both phenomena but conflates mechanisms or omits spin multiplicity change.Incorrect matrix forms (e.g., real σ_y); fails to identify zero-point energy entirely; describes fluorescence and phosphorescence as merely 'fast' vs 'slow' without quantum mechanical basis.
Derivation rigour25%12.5For (a): derives [σ_i, σ_j] = 2iε_ijkσ_k from matrix multiplication or constructs σ_± ladder operators. For (b): full radial equation setup, separation of variables, substitution of trial solution R(r) = exp(-r/na₀)×polynomial, derivation of Bohr radius a₀ = 4πε₀ℏ²/μe², and explicit energy quantization. For (c): explains ISC rate via Fermi's golden rule with spin-orbit coupling matrix element.States commutation relations without derivation; uses known formula for E_n without derivation; describes ISC qualitatively without rate expression.No derivations attempted; merely quotes final results; no mathematical structure for any part.
Diagram / FBD15%7.5For (a): Bloch sphere representation showing spinor |ψ⟩ = cos(θ/2)|↑⟩ + e^{iφ}sin(θ/2)|↓⟩ with θ, φ angles. For (b): radial probability distribution plot 4πr²|R₁₀|² vs r showing peak at a₀. For (c): complete Jablonski diagram with S₀, S₁, S₂, T₁ levels, absorption, fluorescence, phosphorescence arrows, ISC and internal conversion wavy lines, labeled with typical timescales.Sketch of coordinate axes for spin components; generic hydrogen orbital diagram without radial probability; partial energy level diagram missing T₁ or ISC.No diagrams; or incorrect diagrams (e.g., continuous energy spectrum for hydrogen; confusing fluorescence with Raman scattering).
Numerical accuracy15%7.5For (b): E₁ = -13.6 eV = -2.18×10⁻¹⁸ J; Bohr radius a₀ = 0.529 Å = 52.9 pm; zero-point energy identified as +13.6 eV kinetic or ℏ²/2μa₀²; correct reduced mass μ = m_e m_p/(m_e+m_p) ≈ 0.9995m_e with Rydberg constant R_H = 109677 cm⁻¹. For (c): typical fluorescence lifetime 1-10 ns, phosphorescence 1 ms-1 s cited.Correct order of magnitude for energy (-10 eV) and a₀ (~0.5 Å); approximate lifetime ratios without precise values.Energy positive or off by factor of 2; a₀ in wrong units (nm instead of Å); lifetimes incorrect by orders of magnitude.
Physical interpretation20%10For (a): interprets spin as intrinsic angular momentum with no classical analog, connects to Stern-Gerlach experiment quantization. For (b): explains zero-point energy as consequence of position-momentum uncertainty—localized electron requires non-zero kinetic energy; contrasts with classical collapse. For (c): explains phosphorescence forbidden nature via spin selection rule ΔS=0, ISC enabled by heavy atom effect; discusses Indian research context (CCMB fluorescence imaging, IISc OLED phosphorescent materials).States spin is 'quantum property' without Stern-Gerlach; mentions uncertainty principle without explicit ΔxΔp argument; lists applications without mechanistic connection.Classical interpretation of spin as literal rotation; no interpretation of zero-point energy; applications merely listed without scientific rationale.

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