(a) Explain how the uncertainty in position is different from the uncertainty or inaccuracy of the measuring instruments. 10 marks
(b) Determine the ground state energy of an electron in an infinite potential well of width of 2 Å. 10 marks
(c) Draw t

Question

(a) Explain how the uncertainty in position is different from the uncertainty or inaccuracy of the measuring instruments. 10 marks
(b) Determine the ground state energy of an electron in an infinite potential well of width of 2 Å. 10 marks
(c) Draw the normal Zeeman pattern for ¹F₃—¹D₂ transition. 10 marks
(d) In case of pure rotational states, if the temperature will be doubled, then calculate the rotational quantum number corresponding to maximum population density. [Assume that temperature is high] 10 marks
(e) The quantum numbers of two electrons in a two-valence electron atom are n₁ = 6, l₁ = 3, s₁ = ½; n₂ = 5, l₂ = 1, s₂ = ½. Assuming L-S coupling, find the possible values of L and J. 10 marks

UPSC Answer Check · Accepted Answer

Begin with part (a) explaining the fundamental distinction between Heisenberg's intrinsic quantum uncertainty and classical instrumental error, using clear conceptual distinction. For part (b), derive the ground state energy formula for infinite potential well and substitute a = 1 Å (half-width), showing E₁ = h²/(8ma²). Part (c) requires drawing the normal Zeeman triplet pattern with proper spacing and polarization labels. Part (d) involves deriving that J_max ≈ √(kT/2B) - ½, showing J_max increases by √2 when T doubles. Part (e) applies L-S coupling rules: L ranges from |l₁-l₂| to l₁+l₂ (i.e., 2,3,4), S=0 or 1, then J from |L-S| to L+S for each case. Allocate approximately 20% time to each part given equal marks distribution.
- Part (a): Distinguish Heisenberg uncertainty principle (intrinsic, ΔxΔp ≥ ℏ/2) from classical instrumental error (reducible with better apparatus); cite that uncertainty principle holds even with perfect instruments
- Part (b): Ground state energy E₁ = h²/(8ma²) = π²ℏ²/(2mL²) where L = 2a = 2 Å; calculate numerical value ~9.4 eV or 150 aJ
- Part (c): Normal Zeeman effect shows triplet pattern: π component (Δm=0, unshifted) and σ⁺, σ⁻ components (Δm=±1, shifted by ±eℏB/2m); draw energy level diagram with proper spacing and label polarizations
- Part (d): Rotational population N_J ∝ (2J+1)exp[-BJ(J+1)/kT]; find J_max by differentiation, yielding J_max ≈ √(kT/2B) - ½; when T→2T, J_max increases by factor of √2
- Part (e): L-S coupling: L = 2,3,4; S = 0 (singlet) or 1 (triplet); for S=0: J=L so J=2,3,4; for S=1: J ranges from |L-1| to L+1 giving appropriate values for each L

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies that Heisenberg uncertainty is intrinsic and fundamental (part a); proper infinite well boundary conditions (part b); distinguishes normal from anomalous Zeeman effect (part c); correct Boltzmann population distribution for rotors (part d); accurate L-S coupling rules with proper vector addition (part e)	Mixes up uncertainty types or confuses Zeeman effects; minor errors in coupling scheme; understands concepts but with gaps in quantum mechanical foundations	Fundamental misconceptions: treats uncertainty as instrumental error, wrong well formula, anomalous Zeeman drawn as normal, incorrect population formula, or j-j coupling used instead of L-S
Derivation rigour	20%	10	Full derivation of uncertainty principle from commutators or wave packets (a); complete solution of Schrödinger equation with boundary conditions for infinite well (b); systematic derivation of J_max from population maximization (d); clear step-by-step L-S coupling vector addition with triangle rule (e)	States key formulas without full derivation; skips boundary condition justification; presents final results with minimal intermediate steps	No derivations shown; incorrect formulas stated without justification; missing critical steps like boundary conditions or maximization procedure
Diagram / FBD	20%	10	Clear Zeeman energy level diagram showing ¹F₃ and ¹D₂ splitting into 7 and 5 sublevels respectively, with proper g-factor=1 spacing; transitions obeying Δm=0,±1 selection rules; labeled π and σ components with polarization arrows; clean schematic of infinite well wavefunction (optional but adds value)	Basic Zeeman triplet shown but missing sublevel detail or selection rules; diagram present but lacks clarity in labeling	No diagram for part (c); incorrect pattern drawn (e.g., showing anomalous Zeeman splitting); messy or unlabeled diagrams that confuse rather than clarify
Numerical accuracy	20%	10	Part (b): E₁ ≈ 9.4 eV or 1.51×10⁻¹⁷ J with correct unit conversion (a=1×10⁻¹⁰ m); part (d): correct algebraic expression for J_max and √2 factor identified; proper handling of physical constants (ℏ, mₑ, k_B)	Correct formula but arithmetic error; wrong power of 10; correct approach but final numerical value slightly off	Order of magnitude errors; incorrect constants used; no numerical evaluation where required; confusion between width and half-width in well calculation
Physical interpretation	20%	10	Explains why uncertainty principle implies zero-point energy and delocalization (a); relates well energy scale to atomic dimensions and quantum confinement (b); explains polarization selection rules via angular momentum conservation and dipole radiation (c); interprets J_max shift with temperature as thermal population redistribution (d); relates L-S coupling terms to atomic structure and Hund's rules relevance (e)	Some physical insight but limited connection between mathematical results and physical phenomena; minimal interpretation beyond formula statements	Purely mathematical with no physical meaning attached; fails to explain significance of results or connect to observable phenomena

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