(a) An operator P describing the interaction of two spin 1/2 particles is P = a + bσ⃗₁·σ⃗₂, where a and b are constants, and σ⃗₁ and σ⃗₂ are Pauli matrices of the two spins. The total spin angular momentum S⃗ = S⃗₁ + S⃗₂ = 1/2 ℏ(σ⃗₁ + σ⃗₂). Show that

Question

(a) An operator P describing the interaction of two spin 1/2 particles is P = a + bσ⃗₁·σ⃗₂, where a and b are constants, and σ⃗₁ and σ⃗₂ are Pauli matrices of the two spins. The total spin angular momentum S⃗ = S⃗₁ + S⃗₂ = 1/2 ℏ(σ⃗₁ + σ⃗₂). Show that P, S² and Sz can be measured simultaneously. 15 marks
(b) Consider a stream of particles of mass m each moving in the positive x-direction with kinetic energy E towards the potential barrier V(x) = 0 for x ≤ 0, V(x) = 3E/4 for x > 0. Find the fraction of particles reflected at x = 0. 15 marks
(c) Consider the potential V(x) = { 0, 0 < x < a; ∞, elsewhere } (a) Estimate the energies of the ground state as well as those of the first and the second excited states for (i) an electron enclosed in a box of size a = 10⁻¹⁰ m. (ii) a 1 g metallic sphere which is moving in a box of size a = 10 cm. (b) Discuss the importance of the Quantum effects for both of these systems. (c) Estimate the velocities of the electron and the metallic sphere using uncertainty principle. 20 marks

UPSC Answer Check · Accepted Answer

Solve this multi-part quantum mechanics problem by first proving the simultaneous measurability in part (a) through operator commutation, then calculating reflection coefficient in part (b) using boundary conditions, and finally computing energy levels and velocities for both quantum and classical systems in part (c). Allocate approximately 30% time to part (a), 25% to part (b), and 45% to part (c) given its higher mark weightage and computational complexity. Conclude with a comparative discussion highlighting why quantum effects dominate at atomic scales but are negligible for macroscopic objects.
- Part (a): Prove [P, S²] = 0 and [P, Sz] = 0 using σ⃗₁·σ⃗₂ = (S²/ℏ²) - 3/2 and [S², Sz] = 0, establishing complete set of commuting observables
- Part (b): Apply boundary conditions at x=0 for wavefunctions ψ_I = Ae^(ikx) + Be^(-ikx) and ψ_II = Ce^(ik'x) with k=√(2mE)/ℏ, k'=√(2m(E-V))/ℏ=√(2mE/4)/ℏ to find reflection coefficient R = |B/A|² = (k-k')²/(k+k')² = 1/9
- Part (c)(i): Calculate E_n = n²π²ℏ²/(2ma²) for electron (m=9.11×10⁻³¹ kg, a=10⁻¹⁰ m): E₁≈6.02×10⁻¹⁸ J≈37.6 eV, E₂≈150.4 eV, E₃≈338.4 eV
- Part (c)(ii): Calculate same formula for metallic sphere (m=10⁻³ kg, a=0.1 m): E₁≈5.49×10⁻⁶⁴ J, showing extremely closely spaced levels
- Part (c)(b): Discuss quantum effects: electron shows discrete levels, zero-point energy, wave-particle duality; sphere behaves classically with continuous energy spectrum
- Part (c)(c): Apply ΔxΔp≥ℏ/2 to estimate v_electron≈1.1×10⁶ m/s (relativistic correction may be noted) and v_sphere≈5×10⁻³¹ m/s (effectively at rest)
- Demonstrate understanding that quantum-classical correspondence principle requires n→∞ or ℏ→0 limits

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies that simultaneous measurement requires [P,S²]=[P,Sz]=[S²,Sz]=0; recognizes part (b) involves E>V case with partial reflection; distinguishes quantum (discrete) vs classical (continuous) behavior in part (c); correctly applies uncertainty principle for velocity estimation	Shows basic commutation relations but makes algebraic errors; identifies correct physics in barrier problem but confuses E<V tunneling case; calculates energy levels correctly but misses physical interpretation of quantum-classical transition	Confuses simultaneous measurability with uncertainty principle; applies wrong boundary conditions (discontinuity in ψ or dψ/dx); uses classical formulas for quantum systems or vice versa; orders of magnitude errors in numerical estimates
Derivation rigour	20%	10	Rigorous proof using σ⃗₁·σ⃗₂ = 2S₁·S₂/ℏ² = (S²-S₁²-S₂²)/ℏ² = S²/ℏ² - 3/2; systematic application of continuity of ψ and dψ/dx at x=0; clear derivation of E_n = n²h²/(8ma²) with proper handling of infinite potential boundary conditions	States key commutation results without full derivation; applies boundary conditions but with minor algebraic slips; derives energy formula with some steps omitted or uses h instead of ℏ inconsistently	Missing crucial steps in commutation proofs; incorrect boundary conditions (ψ=0 at x=0 for finite barrier); fundamental errors in solving Schrödinger equation; wrong energy level formula (e.g., missing factor of 2 or using wrong mass)
Diagram / FBD	15%	7.5	Clear schematic for part (b) showing incident, reflected, transmitted waves with labeled amplitudes and wave numbers; energy level diagram for part (c) comparing widely spaced electron levels versus impossibly dense sphere levels; sketch of wavefunctions for particle in a box showing nodes	Basic diagram for barrier problem without proper labeling; simple listing of energy values without visual comparison; missing wavefunction sketches or poorly drawn	No diagrams despite physical situations warranting them; incorrect or misleading sketches (e.g., discontinuous wavefunctions, wrong node count); diagrams that contradict written solution
Numerical accuracy	25%	12.5	Precise values: R=1/9≈0.111 or 11.1%; electron E₁≈37.6 eV (or 6.0×10⁻¹⁸ J), E₂≈150.4 eV, E₃≈338.4 eV; sphere energies ~10⁻⁶⁴ J; velocities with correct orders of magnitude and units; proper use of significant figures	Correct formulas with minor calculation errors (within factor of 2-5); correct orders of magnitude but wrong numerical prefactors; inconsistent units (eV vs J) without conversion	Orders of magnitude errors (e.g., electron energy in keV or sphere energy in J); incorrect reflection coefficient (e.g., negative or >1); velocity estimates violating physical bounds (v>c for electron); missing numerical values entirely
Physical interpretation	20%	10	Insightful discussion: explains why triplet/singlet states make P diagonal; interprets partial reflection as wave nature manifestation; clearly demonstrates quantum-classical correspondence—electron shows quantization, zero-point energy, delocalization while 1g sphere has immeasurably small ΔE, negligible v, essentially classical; connects to real systems (atomic spectra vs macroscopic objects)	States that quantum effects are 'important for small systems' without quantitative justification; mentions correspondence principle without demonstrating it; generic statements about wave-particle duality without specific application	Claims quantum effects are equally important for both systems; confuses quantum and classical predictions; no discussion of why we don't observe quantum effects in everyday objects; misses zero-point energy significance

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