Physics 2024 Paper II 50 marks Compulsory Solve

Q1

Q1. (a) A particle limited to the x-axis has the wave function φ(x) = bx² between x = 0 and x = 2; the wave function φ(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1·0 and x = 1·5. (ii) Find the expectation value < x > of the particle position. 10 marks (b) Show that the square of the orbital angular momentum operator (L²) commutes with any of the components of angular momentum operator L. Is it possible to measure L², Lₓ, Lᵧ and Lᵤ simultaneously ? Give reasons for your answer. 6+4=10 marks (c) How is Rydberg constant related to emission wavelength of hydrogen spectrum ? 10 marks (d) Explain how the hydrogen spectrum is used for imaging the universe. 10 marks (e) Find the energy of the particle of mass m moving in a potential field V(x) = 2ℏ²b²x²/m for which the time independent wave function is ψ(x) = exp(– bx²). Here b is a constant. 10 marks

हिंदी में प्रश्न पढ़ें

Q1. (a) एक कण का तरंग फलन φ(x), x-अक्ष में x = 0 और x = 2 के बीच में bx² है एवं अन्य किसी स्थान पर φ(x) = 0 है। (i) x = 1·0 और x = 1·5 के बीच में कण के पाए जाने की प्रायिकता ज्ञात कीजिए। (ii) कण की स्थिति का प्रत्याशा मान < x > ज्ञात कीजिए। 10 अंक (b) दिखाइए कि कक्षक कोणीय संवेग संकारक का वर्ग (L²), कोणीय संवेग संकारक L के किसी भी घटक से दिक्परिवर्तक है। कारण सहित बताइए कि क्या L², Lₓ, Lᵧ और Lᵤ को युगपत स्थिति में मापा जा सकता है। 6+4=10 अंक (c) रिडबर्ग स्थिरांक हाइड्रोजन के स्पेक्ट्रम के उत्सर्जन तरंगदैर्ध्य से किस प्रकार संबंधित है ? 10 अंक (d) व्याख्या कीजिए कि हाइड्रोजन स्पेक्ट्रम किस प्रकार ब्रह्मांड को प्रतिबिंबित करने के लिए उपयोग किया जाता है। 10 अंक (e) उस कण की ऊर्जा ज्ञात कीजिए जिसका द्रव्यमान m है और जो विभव क्षेत्र V(x) = 2ℏ²b²x²/m में गतिमान है और जिसका समय मुक्त तरंग फलन ψ(x) = exp(– bx²) है। यहाँ b एक स्थिरांक है। 10 अंक

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Approach

Solve each sub-part systematically with clear mathematical working: spend ~20% time on (a)(i)-(ii) normalization and probability calculations; ~20% on (b) commutation relations with [L²,Lᵢ]=0 proof; ~20% on (c) Rydberg formula derivation; ~20% on (d) 21-cm line and cosmological applications; ~20% on (e) Schrödinger equation verification. Begin with normalization for (a), state commutation algebra for (b), derive 1/λ = R(1/n₁² - 1/n₂²) for (c), discuss HI regions and redshift for (d), and substitute ψ into TISE for (e).

Key points expected

  • (a)(i) Normalization of φ(x)=bx² on [0,2] to find b, then probability integral P=∫₁^₁·⁵|φ|²dx with correct limits and evaluation
  • (a)(ii) Expectation value ⟨x⟩ = ∫₀² x|φ|²dx / ∫₀²|φ|²dx with proper substitution and algebraic simplification
  • (b) Proof that [L²,Lₓ]=[L²,Lᵧ]=[L²,Lᵤ]=0 using [Lᵢ,Lⱼ]=iℏεᵢⱼₖLₖ and L²=Lₓ²+Lᵧ²+Lᵤ²; explanation that L² commutes with each Lᵢ but [Lₓ,Lᵧ]=iℏLᵤ≠0 prevents simultaneous measurement of all components
  • (c) Derivation of Rydberg formula from Bohr model or quantum mechanics: 1/λ = R_H(1/n₁² - 1/n₂²) with R_H = mₑe⁴/(8ε₀²h³c) = 1.097×10⁷ m⁻¹; mention reduced mass correction
  • (d) Explanation of 21-cm hyperfine transition in neutral hydrogen; mapping galactic spiral arms (Indian astronomers like Radhakrishnan and Gopal-Krishna's work on galactic magnetic fields); cosmological redshift and large-scale structure mapping
  • (e) Substitution of ψ(x)=exp(-bx²) into time-independent Schrödinger equation: -ℏ²/2m · d²ψ/dx² + V(x)ψ = Eψ; calculation of derivatives and verification that E=ℏ²b/m satisfies the equation

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies normalization condition for (a); states all three commutation relations [L²,Lᵢ]=0 for (b); accurately presents Rydberg formula with physical constants for (c); correctly identifies 21-cm line mechanism for (d); properly sets up TISE with kinetic and potential terms for (e)Minor errors in normalization limits or commutation algebra; incomplete Rydberg derivation; vague mention of hydrogen mapping without 21-cm specificity; correct final energy but missing intermediate stepsFundamental misunderstanding of probability interpretation in QM; incorrect commutation relations; confuses Rydberg with other constants; no mention of hyperfine structure; algebraic errors in Schrödinger substitution
Derivation rigour20%10Step-by-step normalization with explicit integral evaluation; complete commutation proof using Levi-Civita symbol or component expansion; full Bohr model derivation to Rydberg formula; systematic TISE verification with all derivatives shownCorrect approach but skips algebraic steps; commutation proof incomplete (shows one component only); Rydberg formula stated without derivation; energy eigenvalue found but derivative steps condensedMissing crucial steps in normalization (no integration shown); asserts commutation without proof; no derivation attempt for Rydberg; incorrect differentiation of Gaussian wave function
Diagram / FBD10%5Clear sketch of wave function φ(x)=bx² on [0,2] with shaded probability region for 1<x<1.5 in (a); vector diagram showing L² as magnitude and Lₓ,Lᵧ,Lᵤ as components with uncertainty sphere for (b); energy level diagram for hydrogen transitions in (c); schematic of 21-cm line emission from HI regions with redshift indication for (d)Basic wave function sketch without probability shading; generic angular momentum diagram; standard hydrogen energy level diagram; simple mention of redshift without diagramNo diagrams where clearly applicable; irrelevant or incorrect diagrams; missing visual representation of probability density
Numerical accuracy25%12.5Exact value b=(105/32)^(1/2) from normalization; probability P=651/1024≈0.636 or exact fraction; ⟨x⟩=16/7≈2.286; correct R_H=1.097×10⁷ m⁻¹; energy eigenvalue E=ℏ²b/m with full algebraic verificationCorrect numerical methods with minor arithmetic errors; approximate values acceptable if method clear; correct final energy with one algebraic slipMajor calculation errors in integrals; incorrect probability outside [0,1]; wrong Rydberg value by orders of magnitude; energy eigenvalue not verified or incorrect
Physical interpretation25%12.5Explains why probability peaks near x=2 for (a); clarifies that L² and one Lᵢ can be simultaneous observables but not all three components due to non-commutation—referencing uncertainty principle; connects Rydberg to atomic spectra fingerprints; discusses how 21-cm line penetrates dust enabling mapping of galactic structure and cosmic dawn (relevant to India's SARAS experiment); interprets Gaussian ψ as ground state of harmonic-like potential with zero nodesBasic interpretation of results without deep physical insight; mentions uncertainty principle superficially; standard description of spectroscopic applications; identifies ground state without discussing whyNo interpretation of mathematical results; fails to explain why simultaneous measurement is limited; no connection to observational astronomy; treats (e) as purely mathematical exercise

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