(a) Find the minimum magnetic field needed for the Zeeman effect to be observed in a spectral line of 400 nm wavelength when a spectrometer whose resolution is 0·010 nm is used. Write the answer in the nearest high integer. 10 marks

(b) Normalised w

Question

(a) Find the minimum magnetic field needed for the Zeeman effect to be observed in a spectral line of 400 nm wavelength when a spectrometer whose resolution is 0·010 nm is used. Write the answer in the nearest high integer. 10 marks

(b) Normalised wave function of hydrogen atom for 1s state is
$$
\psi_{100} = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}, 	ext{ where } a_0 = \frac{\hbar^2}{me^2}
$$
being the Bohr radius. Calculate the expectation value of potential energy in this state. 10 marks

(c) A beam of 12 eV electron is incident on a potential barrier of height 25 eV and width 0·05 nm. Calculate the transmission coefficient. 10 marks

(d) Calculate the Larmor precessional frequency for a magnetic induction field of 0·5 T. Hence calculate the splitting in wave numbers of a spectral line due to normal Zeeman effect for the same field. 10 marks

(e) The first line in the pure rotational spectrum of HCl appears at 21·18 cm⁻¹. Calculate bond length of the molecule. Given atomic masses of H and Cl are 1·008 and 35·45 amu, respectively. 10 marks

UPSC Answer Check · Accepted Answer

Calculate numerical values for all five sub-parts with systematic derivations. Allocate approximately 20% time each to parts (a), (b), (c), (d), and (e) as they carry equal marks. Begin each sub-part by stating the relevant formula, show substitution with proper units, and conclude with the final numerical answer rounded as specified. For part (a), explicitly state the resolution criterion; for (d), show both Larmor frequency and wave number splitting.
- Part (a): Apply the criterion that Zeeman splitting must exceed spectrometer resolution; use Δλ = (eλ²B)/(4πmₑc²) or equivalent relation with Δλ = 0.010 nm to find B_min ≈ 0.43 T → round to 1 T (nearest high integer)
- Part (b): Calculate ⟨V⟩ = ⟨-e²/r⟩ using ψ₁₀₀; evaluate ∫ψ*(-e²/r)ψ d³r = -e²/πa₀³ ∫e^(-2r/a₀)(1/r)4πr²dr = -e²/a₀; final answer -27.2 eV or -2e²/a₀
- Part (c): Apply WKB or exact formula for rectangular barrier: T ≈ exp[-2κL] where κ = √[2m(V₀-E)]/ℏ; with E=12 eV, V₀=25 eV, L=0.05 nm, compute κ and transmission coefficient ≈ 10⁻⁴ to 10⁻⁵
- Part (d): Calculate Larmor frequency ω_L = eB/2mₑ = 4.40×10¹⁰ rad/s (or ν_L = 7.0×10⁹ Hz); then Zeeman splitting Δṽ = eB/4πmₑc = 0.467 cm⁻¹ (normal triplet separation)
- Part (e): Use rigid rotor formula E_J = BJ(J+1) where first line is 2B = 21.18 cm⁻¹; find B = 10.59 cm⁻¹, then I = ℏ/4πcB; calculate reduced mass μ = m_H m_Cl/(m_H+m_Cl); finally r₀ = √(I/μ) ≈ 1.27 Å

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies: resolution-limited Zeeman observation criterion for (a); virial theorem application or direct integration for (b); quantum tunneling through square barrier for (c); Larmor precession and normal Zeeman splitting for (d); rigid rotor energy levels and moment of inertia relation for (e)	Identifies most concepts correctly but confuses normal vs anomalous Zeeman effect, or uses classical instead of quantum expressions for barrier penetration	Misidentifies fundamental concepts—e.g., uses ΔE = μ_BB for (a) without wavelength conversion, treats barrier as classical, or uses vibrational instead of rotational spectrum for (e)
Derivation rigour	20%	10	Shows complete derivations: wavelength-resolution relation leading to B_min for (a); radial integral evaluation with proper Jacobian for (b); WKB or exact transmission derivation for (c); Larmor frequency from torque equation then Δṽ from energy splitting for (d); rotational constant to bond length via reduced mass for (e)	States key formulas correctly but skips intermediate steps or assumes standard results without derivation; minor algebraic gaps	Writes final formulas without derivation, plugs numbers directly, or contains major errors in derivation steps (e.g., wrong volume element in spherical coordinates)
Diagram / FBD	10%	5	Includes clear diagram for (c) showing incident, reflected, transmitted waves with decaying wavefunction in barrier region; and/or energy level diagram for (d) showing normal Zeeman triplet splitting with Δm = 0, ±1 transitions	Sketchy or incomplete diagrams, or diagrams present but not clearly labeled with key parameters (κ, L, V₀, E)	No diagrams where clearly needed (barrier penetration, Zeeman splitting), or completely irrelevant sketches
Numerical accuracy	35%	17.5	All five numerical answers correct with proper significant figures: (a) 1 T (nearest high integer), (b) -27.2 eV or -2e²/a₀, (c) ~3×10⁻⁵ to 10⁻⁴, (d) ω_L ≈ 4.4×10¹⁰ rad/s and Δṽ ≈ 0.47 cm⁻¹, (e) r₀ ≈ 1.27 Å; consistent unit conversions throughout	Correct order of magnitude for most answers but arithmetic errors in 1-2 parts; correct final answer for (a) but wrong rounding convention	Major numerical errors in multiple parts; wrong powers of 10; failure to convert units (nm to m, cm⁻¹ to m⁻¹, amu to kg); incorrect rounding for (a)
Physical interpretation	15%	7.5	Interprets (a) why resolution limits observable splitting; (b) relates ⟨V⟩ = 2E₁ to virial theorem; (c) explains exponential sensitivity to barrier width; (d) connects Larmor frequency to observable spectral line splitting; (e) validates bond length against known HCl value (~1.27 Å)	Brief interpretation for 2-3 parts but superficial; states results without explaining physical significance	No physical interpretation provided; fails to check reasonableness of answers; absurd results (e.g., B ~ 10⁵ T, bond length ~10⁻¹⁵ m) not flagged

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