(a) What is de Broglie concept of matter wave ? Evaluate de Broglie wavelength of Helium that is accelerated through 300 V.
(Given mass of proton = Mass of neutron = 1·67×10⁻²⁷ kg) 10 marks

(b) An electron in a one-dimensional infinite potential wel

Question

(a) What is de Broglie concept of matter wave ? Evaluate de Broglie wavelength of Helium that is accelerated through 300 V.
(Given mass of proton = Mass of neutron = 1·67×10⁻²⁷ kg) 10 marks

(b) An electron in a one-dimensional infinite potential well, defined by
V(x) = 0 for -a ≤ x ≤ a and V(x) = ∞
otherwise, goes from n = 4 to n = 2 level and emits photon of frequency 3·43×10¹⁴ Hz. Calculate the width of the well. (Assume Plank's constant h = 6·626×10⁻³⁴ J.S. and mass of electron m = 9·11×10⁻³¹ kg) 10 marks

(c) Calculate the magnetic field strength required to observe the NMR spectrum of protons in benzene at 120 MHz. [Given the value of nuclear g-factor gₙ for protons is 5·585] 10 marks

(d) Show that the Landé g-factor for pure orbital angular momentum and pure spin angular momentum are 1 and 2 respectively. Further, evaluate the g-factor for the state ³P₁. 10 marks

(e) The raising (J₊) and lowering (J₋) operators are defined by J₊ = Jₓ + iJᵧ and J₋ = Jₓ - iJᵧ respectively. Prove the following identities :
(i) [Jᵤ, J₊] = ±ℏJ₊
(ii) J₋J₊ = J² - Jᵤ² - ℏJᵤ 10 marks

UPSC Answer Check · Accepted Answer

Solve requires systematic problem-solving across all five sub-parts with equal time allocation (~20% each). Begin with concise conceptual definitions for (a), then proceed to step-by-step calculations for (a)-(c), rigorous derivations for (d)-(e). Structure: direct answers without elaborate introduction, showing all intermediate steps, unit conversions, and final boxed results for each sub-part.
- (a) State de Broglie hypothesis λ = h/p; derive λ = h/√(2mE) for non-relativistic case; calculate He wavelength using m_He = 4m_p (alpha particle mass) and E = 300 eV, obtaining λ ≈ 8.3×10⁻¹² m
- (b) Apply infinite square well energy levels E_n = n²π²ℏ²/(8ma²) for well width 2a; use ΔE = E₄ - E₂ = hf to solve for a, obtaining a ≈ 1.5×10⁻⁹ m or width 2a ≈ 3 nm
- (c) Apply NMR resonance condition hν = gₙμₙB where μₙ = eℏ/(2m_p); solve for B = hν/(gₙμₙ) ≈ 2.82 T
- (d) Derive g_L = 1 from μ_L = -(e/2m)L and g_S = 2 from μ_S = -(e/m)S; apply Landé formula g_J = 1 + [J(J+1)+S(S+1)-L(L+1)]/[2J(J+1)] for ³P₁ (L=1,S=1,J=1) to get g_J = 3/2
- (e) Prove [J_z, J_+] = ℏJ_+ and [J_z, J_-] = -ℏJ_- using [J_z, J_x] = iℏJ_y and [J_z, J_y] = -iℏJ_x; prove J_-J_+ = J² - J_z² - ℏJ_z by expanding and using J_x² + J_y² = J² - J_z²

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies de Broglie matter-wave duality, infinite potential well boundary conditions, NMR resonance physics, Landé factor physical basis, and angular momentum operator algebra; no conceptual errors across any sub-part	Minor errors in one sub-part such as confusing well width with half-width, or misidentifying quantum numbers in Landé formula; core concepts mostly correct	Fundamental misconceptions like treating de Broglie wavelength as electromagnetic wave, or incorrect commutation relations in operator algebra
Derivation rigour	20%	10	Complete step-by-step derivations: energy-momentum relation for thermal de Broglie wavelength, quantization condition for infinite well, explicit commutation algebra for (e); all steps logically connected with proper justification	Derivations mostly complete but skips key steps like justifying non-relativistic approximation or algebraic simplifications in operator proofs; final results correct but reasoning gaps	Missing derivations entirely or circular reasoning; states formulas without derivation where required; mathematically invalid steps like incorrect operator expansion
Diagram / FBD	10%	5	Clear diagram for (b) showing infinite potential well with labeled width 2a, energy levels n=2 and n=4 with wavefunction nodes indicated; schematic for NMR precession in (c)	Basic well sketch without energy level labeling or node structure; diagram present but lacks clarity in representing quantum mechanical features	No diagrams where helpful, or misleading sketches showing classical particle trajectories instead of probability distributions
Numerical accuracy	30%	15	All three calculations (a, b, c) correct within significant figures: λ_He ≈ 8.3×10⁻¹² m or 8.2×10⁻¹² m (using exact values), well width ≈ 3.0-3.2 nm, B ≈ 2.8 T; proper unit handling and scientific notation	Correct method but arithmetic errors or unit conversion mistakes (e.g., eV to J, MHz to Hz); order of magnitude correct but final digit errors	Wrong formulas leading to incorrect orders of magnitude; missing factors of 2 or π; no unit consistency checks
Physical interpretation	20%	10	Interprets de Broglie wavelength significance for electron microscopy resolution; connects well width to nanoscale semiconductor quantum dots; explains g-factor deviation from 1 or 2 as spin-orbit coupling evidence; relates raising/lowering operators to angular momentum quantization	Brief mention of physical significance without elaboration; standard concluding statements without specific context to atomic physics applications	Purely mathematical manipulation with no physical insight; fails to interpret what calculated values mean for real systems

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