Physics 2021 Paper II 50 marks Calculate

Q3

(a) In observing the Raman spectrum of a sample using 3637Å as the exciting line, one gets stoke line at 3980Å. Deduce the Raman shift in m⁻¹ units. Compute the wavelength in Å for corresponding stokes and antistokes lines if the exciting line is 6465Å. (20 marks) (b) Explain spin-orbit coupling. Discuss the splitting of spectral lines of H-atom due to spin-orbit coupling. (15 marks) (c) The quantum numbers of two electrons in a two valence electron atom are; n₁=8 l₁=4 s₁=½ n₂=7 l₂=2 s₂=½ (i) Assuming L-S coupling, find the possible value of L and hence of J. (ii) Assuming j-j coupling, find the possible values of J. (7+8 marks)

हिंदी में प्रश्न पढ़ें

(a) 3637Å के उत्तेजन रेखा के रूप में उपयोग करते हुए एक नमूने के रमन वर्णक्रम (स्पेक्ट्रम) को देखने में 3980Å पर स्टोक्स रेखा मिलती है । मीटर⁻¹ इकाई में रमन विस्थापन (शिफ्ट) का पता लगाइए । संबंधित स्टोक्स और एंटी-स्टोक्स लाइनों के लिए Å में तरंग दैर्घ्य की गणना कीजिए यदि उत्तेजन रेखा 6465Å है । (20 अंक) (b) प्रचक्रण-कक्षा युग्मन की व्याख्या कीजिए । प्रचक्रण-कक्षा युग्मन के कारण हाइड्रोजन-परमाणु की स्पेक्ट्रमी (वर्णक्रम) रेखाओं के विपाटन की चर्चा कीजिए । (15 अंक) (c) एक दो संयोजकता वाले इलेक्ट्रॉन परमाणु में दोनों इलेक्ट्रॉनों की क्वांटम संख्याएँ हैं; n₁=8 l₁=4 s₁=½ n₂=7 l₂=2 s₂=½ (i) L-S युग्मन को मानते हुए L का संभावित मान ज्ञात कीजिए और J का भी । (ii) j-j युग्मन को मानते हुए J का संभावित मान ज्ञात कीजिए । (7+8 अंक)

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Approach

This is a calculation-heavy question demanding precise numerical work for (a) and (c), plus conceptual explanation for (b). Allocate approximately 40% time to part (a) for Raman shift calculations and wavelength conversions, 30% to part (b) for explaining spin-orbit coupling mechanism and H-atom fine structure, and 30% to part (c) for L-S and j-j coupling term calculations. Begin with clear statements of formulas, show all intermediate steps, and conclude with physical significance of results.

Key points expected

Part (a): Correct application of Raman shift formula Δν̃ = (1/λ₀ - 1/λs) in m⁻¹; calculation of Stokes and anti-Stokes wavelengths for new exciting line using the same Raman shift
Part (b): Explanation of spin-orbit coupling as interaction between electron's spin magnetic moment and orbital magnetic field; derivation of fine structure splitting in H-atom showing ΔE ∝ j(j+1) - l(l+1) - s(s+1)
Part (c)(i): L-S coupling: L = |l₁ - l₂| to l₁ + l₂ = 2,3,4,5,6; S = 0,1; J values from |L-S| to L+S for each combination
Part (c)(ii): j-j coupling: j₁ = l₁ ± ½ = 7/2, 9/2; j₂ = l₂ ± ½ = 3/2, 5/2; J from |j₁-j₂| to j₁+j₂
Physical interpretation: Term symbols notation ²ˢ⁺¹L_J for L-S coupling; comparison of coupling schemes showing L-S dominates for light atoms, j-j for heavy atoms

Evaluation rubric

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	Correctly identifies Raman scattering as inelastic with frequency shift independent of incident wavelength; accurately describes spin-orbit Hamiltonian H_SO = ξ(r)L·S; properly distinguishes L-S coupling (weak spin-orbit, strong electrostatic) from j-j coupling (strong spin-orbit, weak residual electrostatic) with correct applicability domains	Basic understanding of Raman effect and spin-orbit coupling present but confuses Stokes/anti-Stokes selection rules or misidentifies which coupling scheme applies to which atomic regime; minor errors in quantum number addition rules	Fundamental misconceptions such as treating Raman shift as wavelength difference rather than wavenumber difference; confuses spin-orbit coupling with Zeeman effect; applies wrong coupling scheme or violates Pauli exclusion principles in term construction
Derivation rigour	20%	10	Shows complete derivation for Raman shift conservation: Δν̃ = constant implies 1/λs' = 1/λ₀' - Δν̃; derives spin-orbit splitting energy for H-atom using perturbation theory with radial integral ξ_n,l; systematically enumerates all L-S and j-j coupling states using proper vector addition	States key formulas correctly but skips intermediate algebraic steps; presents correct final expressions for J values but shows incomplete working for intermediate quantum number combinations	Jumps to answers without showing derivation; algebraic errors in reciprocal wavelength calculations; incorrect application of triangle rules for angular momentum coupling; missing steps in perturbation derivation
Diagram / FBD	10%	5	Clear energy level diagram showing fine structure splitting of H-atom n=2 level with ²P₁/₂ and ²P₃/₂ states; schematic of Raman process illustrating virtual level transitions; term diagram showing L-S coupling hierarchy	Basic level diagram present but missing labels or incorrect spacing; simple schematic of scattering process without energy scale	No diagrams provided where clearly needed; or diagrams with major errors such as wrong number of split levels, incorrect degeneracy labeling
Numerical accuracy	30%	15	Precise calculation: Raman shift = (1/3637×10⁻¹⁰ - 1/3980×10⁻¹⁰) = 2.37×10⁴ m⁻¹; new Stokes λ = 7065Å, anti-Stokes λ = 5968Å; all J values correctly enumerated with correct multiplicities for both coupling schemes	Correct method but arithmetic errors in wavelength conversion; minor errors in counting J values or missing some combinations; correct order of magnitude for Raman shift	Major calculation errors such as using wavelength difference instead of reciprocal; wrong powers of ten; incorrect J value ranges; fails to convert units properly between Å and m
Physical interpretation	15%	7.5	Explains why Raman shift is material-specific and independent of excitation wavelength; connects spin-orbit splitting to relativistic origin and fine structure constant α; discusses how L-S to j-j coupling transition occurs with increasing Z (e.g., Pb, Hg vs. C, N, O); mentions selection rules ΔJ = 0, ±1 for transitions	Brief mention of physical significance without elaboration; states that heavier atoms show j-j coupling without explaining why; mentions fine structure but not its relation to Dirac theory	No physical interpretation provided; purely mathematical answer; fails to explain observable consequences or experimental relevance

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