(a) Find out the difference in frequencies of Lyman-alpha line in hydrogen and deuterium atoms. 15 marks

(b) The Stern-Gerlach experiment is a landmark experiment in quantum mechanics. Discuss about the most important findings of this experiment. 15

Question

(a) Find out the difference in frequencies of Lyman-alpha line in hydrogen and deuterium atoms. 15 marks

(b) The Stern-Gerlach experiment is a landmark experiment in quantum mechanics. Discuss about the most important findings of this experiment. 15 marks

(c) (i) From the pure rotational absorption spectra of a diatomic molecule (HF), the wave number difference between the consecutive rotational lines is found to be Δν̄ = 4050 m⁻¹. Calculate the following:
(1) Rotational constant
(2) Moment of inertia
(3) Distance between two atoms (bond length)
[Given, M_H = 1 u, M_F = 19 u] 10 marks

(ii) The force constant of HCl molecule is 4.8×10⁵ dyne/cm. Calculate the wave numbers of Stokes and anti-Stokes lines, when excited with a radiation of wavelength 4358 Å.
[Given, μ_HCl = 1.61×10⁻²⁴ g] 10 marks

UPSC Answer Check · Accepted Answer

Begin with the directive to calculate and discuss across four sub-parts: spend ~30% time on (a) isotope shift calculation using reduced mass correction; ~25% on (b) discussing Stern-Gerlach findings with experimental schematic; ~25% on (c)(i) rotational spectroscopy of HF; and ~20% on (c)(ii) Raman spectroscopy of HCl. Structure as: brief theory → step-by-step derivation → numerical substitution → final result with units → physical significance.
- (a) Reduced mass calculation for H (μ_H) and D (μ_D), Rydberg formula with reduced mass correction, frequency difference Δν = ν_H − ν_D ≈ 4.53×10¹¹ Hz or equivalent
- (b) Experimental setup with inhomogeneous magnetic field, silver atom beam splitting into two discrete components, direct evidence of space quantization and electron spin (intrinsic angular momentum ℏ/2)
- (c)(i) Rotational constant B = Δν̄/2 = 2025 m⁻¹, moment of inertia I = h/(8π²cB), bond length r₀ = √(I/μ) ≈ 0.92 Å for HF
- (c)(ii) Vibrational frequency ω = (1/2π)√(k/μ), Raman shift Δν̄ = ±(ν₀ ∓ ν_vib), Stokes and anti-Stokes lines at ν̄₀ − ν̄_vib and ν̄₀ + ν̄_vib respectively
- Proper unit conversions throughout: CGS to SI for (c)(ii), unified atomic mass to kg, wavenumber to frequency where needed
- Physical significance: isotope shift tests QED predictions, Stern-Gerlach validates quantum mechanics vs classical expectations, spectroscopic constants determine molecular structure

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies reduced mass dependence in (a), space quantization and spin-½ in (b), rigid rotor model in (c)(i), and Raman scattering mechanism with selection rules in (c)(ii); no conceptual errors in any sub-part	Minor errors like using atomic mass instead of reduced mass in (a), confusing orbital and spin angular momentum in (b), or misapplying vibrational frequency formula in (c)(ii)	Fundamental misconceptions such as ignoring reduced mass entirely, attributing Stern-Gerlach splitting to orbital angular momentum only, or treating Raman as fluorescence process
Derivation rigour	20%	10	Complete step-by-step derivations: reduced mass substitution in Rydberg formula, force equation for beam deflection, rotational energy levels J(J+1)ħ²/2I, and vibrational frequency from classical harmonic oscillator; all steps logically connected	Correct final formulas but skips key steps like showing how Δν̄ = 2B arises from selection rule ΔJ = ±1, or assumes Raman shift without deriving from polarizability	Plugs numbers without showing any derivation, or presents garbled derivations with dimensional inconsistencies and missing physical justification
Diagram / FBD	15%	7.5	Clear labeled diagram for Stern-Gerlach apparatus in (b) showing magnet geometry, beam source, screen with doublet splitting; energy level diagrams for rotational transitions in (c)(i) and Raman process in (c)(ii)	Basic Stern-Gerlach sketch without labels or missing beam path; no energy level diagrams for spectroscopic parts	No diagrams despite (b) requiring experimental visualization, or completely incorrect schematic showing continuous beam spreading
Numerical accuracy	25%	12.5	Precise calculations: Δν ≈ 4.5×10¹¹ Hz for (a), B = 2025 m⁻¹, I ≈ 1.37×10⁻⁴⁷ kg·m², r₀ ≈ 0.917 Å for HF; ν̄_vib ≈ 2885 cm⁻¹, Stokes at ~4167 cm⁻¹ and anti-Stokes at ~4550 cm⁻¹ for HCl; proper significant figures and unit handling	Correct method but arithmetic errors like wrong powers of 10, mixing CGS-SI units without conversion, or bond length off by factor of √2 due to reduced mass error	Order-of-magnitude errors, missing unit conversions (especially dyne/cm to N/m), or nonsensical results like bond length in meters without Å conversion
Physical interpretation	20%	10	Interprets (a) as test of nuclear motion effects and QED; (b) as definitive proof of quantization and electron spin; (c)(i)-(ii) as structural determination tools; connects to Indian research context (e.g., Raman effect discovery by C.V. Raman at IISc, 1930 Nobel)	States results without explaining why they matter; mentions Raman effect without acknowledging Indian scientific heritage	Purely mathematical treatment with no physical insight; fails to distinguish spectroscopic techniques or their applications

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