Physics 2021 Paper II 50 marks Prove

Q2

(a) Using Pauli spin matrices prove that, (i) σₓσᵧ + σᵧσₓ = 0; σᵧσᵤ + σᵤσᵧ = 0; σₓσᵤ + σᵤσₓ = 0 (ii) σ₊σ₋ = 2(1+σᵤ) (iii) σₐ + σᵦ = iσᵧ where α ≠ β ≠ γ 8+6+6 marks (b) Find the uncertainty in the momentum of a particle when its position is determined within 0·02 cm. Find also the uncertainty in the velocity of an electron and α-particle respectively when they are located within 15×10⁻⁸ cm. 15 marks (c) A particle is moving in a one dimensional box of width 50Å and infinite height. Calculate the probability of finding the particle within an interval of 15Å at the centres of the box when it is in its state of least energy. 15 marks

हिंदी में प्रश्न पढ़ें

(a) पाउली प्रचक्रण आव्यूहों का उपयोग करते हुए सिद्ध कीजिए कि, (i) σₓσᵧ + σᵧσₓ = 0; σᵧσᵤ + σᵤσᵧ = 0; σₓσᵤ + σᵤσₓ = 0 (ii) σ₊σ₋ = 2(1+σᵤ) (iii) σₐ + σᵦ = iσᵧ जहाँ α ≠ β ≠ γ 8+6+6 अंक (b) एक कण के संवेग में अनिश्चितता का पता लगाइए जब उसकी स्थिति 0·02 cm के भीतर निर्धारित की जाती है। एक इलेक्ट्रॉन और अल्फा कण के वेग में अनिश्चितता का पता लगाइए जब वे 15×10⁻⁸ cm के भीतर स्थित हों। 15 अंक (c) एक कण 50Å चौड़ाई और अनंत ऊँचाई के एकविमीय कोष (बाक्स) में घूम रहा है। कोष (बाक्स) के केंद्र पर 15Å के अंतराल के भीतर कण को खोजने की संभावना (प्रायिकता) की गणना कीजिए जब वह अपनी न्यूनतम ऊर्जा की स्थिति में हो। 15 अंक

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Directive word: Prove

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How this answer will be evaluated

Approach

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.

Key points expected

  • 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

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies Pauli matrix algebra, anti-commutation vs commutation distinctions, ladder operator definitions, Heisenberg principle formulation with ℏ/2 not ℏ, and infinite square well boundary conditions with proper normalizationStates Pauli matrices correctly but confuses commutation with anti-commutation; uses ΔpΔx~ℏ without factor of 2; sets up wavefunction but makes errors in probability interval limitsWrong matrix definitions; treats σ matrices as scalars; uses classical uncertainty concepts; applies wrong potential (finite well or free particle) for part (c)
Derivation rigour25%12.5Step-by-step matrix multiplication shown explicitly for all three proofs in (a); clear algebraic manipulation for σ₊σ₋ expansion; systematic use of εᵢⱼₖ for cyclic relation; all logical steps justifiedSkips intermediate steps in matrix multiplication; assumes results without proof; correct final answers but missing crucial steps in derivation chainNo derivation shown—only states final results; circular reasoning; mathematically invalid steps like dividing by zero or incorrect matrix multiplication order
Diagram / FBD5%2.5Clear sketch of infinite square well potential with walls at x=0 and x=L, marking the interval 15Å at center; optional Bloch sphere representation for spin statesRough sketch of potential well without labeled axes or dimensions; no spin visualizationNo diagrams where helpful; or irrelevant diagrams that don't illuminate the physics
Numerical accuracy30%15Correct unit conversions (cm→m, Å→m); accurate calculation of Δp=2.64×10⁻³¹ kg·m/s for part (b); electron Δv≈2.9×10⁵ m/s, α-particle Δv≈40 m/s; part (c) probability ≈0.498 using exact integration of sin²(πx/L)Correct formulas but arithmetic errors; wrong powers of 10 in unit conversion; approximate probability without exact integrationMajor unit errors (using cm directly); orders of magnitude wrong; incorrect mass values; probability >1 or negative; missing numerical answers entirely
Physical interpretation20%10Explains significance of anti-commutation for spin-½ fermions and Pauli exclusion; interprets Δv results showing quantum effects dominate for electrons vs classical behavior for α-particles; explains why probability at center approaches 0.5 for ground stateBrief mention of quantum-classical distinction without elaboration; states results without physical insightNo interpretation provided; or physically nonsensical statements like 'electron is at rest' or 'probability is exact position'

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