Physics 2022 Paper II 50 marks Explain

Q3

(a) What is the spin wave function (for $s=\frac{1}{2}$) if the spin component in the direction of unit vector $\eta$ has a value of $\frac{1}{2}\hbar$ ? 15 marks (b) (i) Why does Stern-Gerlach experiment enjoy so much importance in atomic physics ? (ii) Draw the schematic diagram of this experiment and comment on the shapes of the magnet pole pieces. (iii) Why was the atomic beam of silver used in this experiment ? 20 marks (c) Define Franck-Condon principle. How does it help in explaining the intensity distribution of vibrational-electronic spectra of diatomic molecules. 15 marks

हिंदी में प्रश्न पढ़ें

(a) यदि इकाई सदिश $\eta$ की दिशा में स्पिन घटक का मान $\frac{1}{2}\hbar$ है तो $s=\frac{1}{2}$ के लिए स्पिन तरंग फलन क्या है ? 15 अंक (b) (i) परमाणु भौतिकी में स्टर्न-गर्लाच प्रयोग का इतना महत्व क्यों है ? (ii) इस प्रयोग का योजनाबद्ध आरेख बनाइए और चुंबक के ध्रुवीय खंडों की आकृतियों पर टिप्पणी कीजिए । (iii) इस प्रयोग में चांदी के परमाणु पुंज का प्रयोग क्यों किया गया था ? 20 अंक (c) फ्रांक-कॉन्डन सिद्धांत को परिभाषित कीजिए । यह द्विपरमाणुक अणुओं के कंपनिक और इलेक्ट्रॉनिक स्पेक्ट्रमों के तीव्रता वितरण को समझाने में कैसे मदद करता है । 15 अंक

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Directive word: Explain

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How this answer will be evaluated

Approach

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.

Key points expected

  • 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

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5Correctly identifies spin-1/2 eigenstates in arbitrary direction; accurately states Stern-Gerlach historical significance including space quantization and spin discovery; precisely defines Franck-Condon principle with quantum mechanical justification; no conceptual errors in angular momentum algebra or selection rulesBasic understanding of spinors but errors in phase factors or rotation angles; partial recognition of Stern-Gerlach importance; vague statement of Franck-Condon principle without quantum mechanical basis; minor confusion between orbital and spin angular momentumFundamental misunderstanding of spin quantization direction; confuses Stern-Gerlach with Zeeman effect; incorrect statement of Franck-Condon principle as energy conservation; serious errors in quantum number assignments
Derivation rigour20%10Complete derivation of spin wave function using S·η = Sx sinθcosφ + Sy sinθsinφ + Sz cosθ with Pauli matrices; explicit eigenvalue solution showing normalization; proper treatment of Stern-Gerlach force F = ∇(μ·B) with magnetic moment derivation; mathematical justification of Franck-Condon overlap integralsPartial derivation with correct final form but skipped steps; correct force expression but incomplete magnetic moment derivation; mentions overlap integrals without explicit calculation; some algebraic errors but correct physical conclusionMissing essential derivation steps or incorrect operator identification; wrong force expression or missing ∇B term; no mathematical basis for Franck-Condon principle; derivation fundamentally flawed or absent
Diagram / FBD15%7.5Clear Stern-Gerlach schematic with labeled components (oven, collimator, magnet, detector); accurate depiction of asymmetric pole pieces (wedge/knife-edge profile) creating field gradient; Franck-Condon diagram showing displaced harmonic oscillator potentials with vertical transition arrow and vibrational levels; proper labeling of axes and parametersBasic Stern-Gerlach diagram missing some labels or with symmetric pole pieces; simplified Franck-Condon curves without proper displacement; diagrams understandable but lacking precision in representing physical featuresMissing or highly inaccurate diagrams; symmetric pole pieces showing no field gradient; no Franck-Condon potential curves; diagrams that misrepresent the physics or are illegible
Numerical accuracy15%7.5Correct numerical factors in spinor coefficients (cos(θ/2), sin(θ/2)); accurate eigenvalue ±ℏ/2; proper magnitude of silver magnetic moment (1 Bohr magneton); correct calculation of force and beam separation estimates; accurate vibrational quantum numbers in Franck-Condon examplesMinor errors in angular factors or missing factor of 2; correct order of magnitude for physical quantities; approximate treatment of vibrational overlap without specific quantum numbersSerious numerical errors in spinor components; wrong eigenvalues; incorrect magnetic moment magnitude; no quantitative treatment where expected; numerical values that contradict physical reality
Physical interpretation25%12.5Clear physical meaning of spin quantization in arbitrary direction; insightful discussion of Stern-Gerlach as paradigm for quantum measurement, state preparation, and decoherence; explanation of why shaped pole pieces create necessary field inhomogeneity; deep understanding of Franck-Condon factors determining spectral intensities and bond length changes; connects to modern applications (quantum computing, molecular spectroscopy)Basic physical interpretation without deeper insight; standard explanation of beam splitting; superficial comment on pole piece shapes; limited discussion of intensity distribution implicationsMissing physical interpretation of mathematical results; no explanation of why experiment matters; inability to explain pole piece geometry purpose; no connection between Franck-Condon principle and observed spectra

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