(a) A particle constrained to move along x-axis in the domain 0 ≤ x ≤ L has a wave function ψ(x) = sin(nπx/L), where n is an integer. Normalize the wave function and evaluate the expectation value of momentum of the particle. (15 marks)

(b) Evaluate

Question

(a) A particle constrained to move along x-axis in the domain 0 ≤ x ≤ L has a wave function ψ(x) = sin(nπx/L), where n is an integer. Normalize the wave function and evaluate the expectation value of momentum of the particle. (15 marks)

(b) Evaluate the most probable distance of the electron of the hydrogen atom in its 2p state. What is the radial probability density at that distance ? (15 marks)

(c) What is nuclear magnetic resonance ? Explain its working principle and use in magnetic resonance imaging systems. (5+5+10=20 marks)

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical-cum-descriptive question by allocating approximately 30% time to part (a) on particle-in-a-box normalization and momentum expectation, 30% to part (b) on hydrogen 2p state radial probability, and 40% to part (c) on NMR principles and MRI applications. Begin each part with the relevant formula, show complete derivation steps, and conclude with physical interpretation—especially connecting MRI to healthcare applications in Indian context like AIIMS Delhi's advanced imaging facilities.
- Part (a): Normalization constant A = √(2/L) obtained by integrating |ψ|² from 0 to L; momentum expectation value ⟨p⟩ = 0 shown via direct integration or operator method
- Part (a): Recognition that ⟨p⟩ = 0 reflects stationary state with equal probability of left/right motion, or explicit calculation using p̂ = -iℏ(d/dx)
- Part (b): Radial wave function R₂₁(r) ∝ r·exp(-r/2a₀) for 2p state; radial probability density P(r) = r²|R₂₁|² ∝ r⁴exp(-r/a₀)
- Part (b): Most probable distance r_mp = 4a₀ obtained by dP/dr = 0; maximum probability density value P(r_mp) = (1/24a₀)·(4/e)⁴ or equivalent simplified form
- Part (c): NMR defined as resonant absorption of RF radiation by nuclear spins in magnetic field; working principle involving Zeeman splitting, Larmor precession, and resonance condition ω = γB₀
- Part (c): MRI working: gradient coils for spatial encoding, RF pulses for excitation, detection of FID signals; T₁/T₂ contrast for tissue differentiation; Indian relevance: indigenous MRI development at BARC, widespread diagnostic use for cancer and neurological disorders

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies that part (a) requires normalization via ∫\|ψ\|²dx=1 and that ⟨p⟩=0 for real stationary states; for (b) uses correct 2p radial function and distinguishes radial probability from probability density; for (c) accurately describes nuclear spin states, resonance condition, and MRI signal generation without confusing with electron spin resonance	Minor errors in normalization limits or confuses \|ψ\|² with ψ; uses approximate 2p form or misses r² factor in radial probability; describes NMR qualitatively but confuses T₁/T₂ or omits gradient field role in MRI	Fundamental misunderstanding: treats momentum expectation as nonzero without calculation, uses wrong principal quantum number for 2p state, or describes MRI as X-ray based imaging
Derivation rigour	20%	10	Complete step-by-step derivations: for (a) explicit integration with limits, momentum operator application showing hermiticity; for (b) clear differentiation of r⁴exp(-r/a₀) with chain rule; for (c) logical flow from nuclear magnetic moment to Bloch equations or signal detection physics	Skips intermediate steps but key results correct; omits justification for boundary conditions or assumes standard results without derivation; descriptive part (c) lacks systematic physics progression	Missing critical steps: no integration shown for normalization, asserts ⟨p⟩=0 without calculation, or presents memorized final answers without any derivation
Diagram / FBD	15%	7.5	For (a): sketches infinite potential well with wave functions; for (b): plots radial probability distribution P(r) showing node structure and peak at 4a₀; for (c): clear NMR energy level diagram with Zeeman splitting, and/or MRI block diagram showing magnet, gradients, RF coil, and patient positioning	At least one relevant diagram present but missing labels or incorrect scales; rough sketches without proper axes; part (c) has generic MRI scanner image without physics annotation	No diagrams despite clear visualization opportunities; or completely incorrect diagrams (e.g., wrong potential shape, confusing 1s and 2p radial functions)
Numerical accuracy	25%	12.5	Exact symbolic results: A = √(2/L), ⟨p⟩ = 0; r_mp = 4a₀ with correct probability density P_max = (128/81a₀)exp(-4) or simplified equivalent; numerical values in terms of a₀, no calculation errors in exponents or coefficients	Correct final expressions but arithmetic slips in intermediate steps; correct r_mp but wrong numerical coefficient in P(r); mixes up 2p with 2s radial function leading to r_mp = 2a₀	Major numerical errors: wrong normalization constant, nonzero ⟨p⟩ due to integration error, or r_mp = a₀ (confusing with Bohr radius as most probable distance for 1s)
Physical interpretation	20%	10	For (a): explains ⟨p⟩=0 as equal probability flux in both directions, connects to classical standing wave; for (b): interprets 4a₀ as larger orbital radius for excited state, contrasts with 1s; for (c): explains T₁/T₂ weighting for different tissues, mentions Indian healthcare applications (early cancer detection, brain imaging), and notes safety advantages over ionizing radiation	Brief mention of physical meaning without elaboration; generic statement that MRI is 'useful for diagnosis' without explaining contrast mechanism or specific applications	No physical interpretation provided; or incorrect interpretation (e.g., claims ⟨p⟩=0 means particle is at rest, or that NMR involves nuclear transitions like gamma decay)

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