Physics 2024 Paper II 50 marks Prove

Q2

Q2. (a) Prove that : (i) [L², Lz] = 0 (ii) [Lz, L+] = ℏL+ (iii) [L+, L-] = 2ℏLz (iv) L+ L- = L² - Lz² + ℏLz where ℏ = h/2π (ℏ is Planck's constant) 5+5+5+5=20 marks (b) The ground state wave function of a harmonic oscillator is ψ₀(x) = (mω/ℏπ)^(1/4) exp(-mωx²/2ℏ). (i) At which point is the probability density maximum ? (ii) What is the value of the maximum probability density ? 15 marks (c) (i) Assuming the potential seen by a neutron in a nucleus to be schematically represented by a one-dimensional, infinite rigid wall potential of length 10⁻¹⁵ m, estimate the minimum kinetic energy of the electron. (ii) Estimate the minimum kinetic energy of neutron bound within the nucleus as described above. Can an electron be confined in a nucleus ? Explain. 15 marks

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Q2. (a) सिद्ध कीजिए कि : (i) [L², Lz] = 0 (ii) [Lz, L+] = ℏL+ (iii) [L+, L-] = 2ℏLz (iv) L+ L- = L² - Lz² + ℏLz जहाँ ℏ = h/2π (h प्लांक स्थिरांक है) 5+5+5+5=20 अंक (b) आध (निम्नतम) अवस्था में एक सरल आवर्ती (सनादि) दोलक का तरंग फलन ψ₀(x) = (mω/ℏπ)^(1/4) exp(-mωx²/2ℏ) है। (i) इसके किस बिंदु पर प्रायिकता घनत्व अधिकतम है ? (ii) अधिकतम प्रायिकता घनत्व का मान क्या है ? 15 अंक (c) (i) यह मानते हुए कि नाभिक में न्यूट्रॉन द्वारा अनुभव किए गए विभव को 10⁻¹⁵ मी. लंबाई के एक-आयामी, अनंत दृढ़ दीवार विभव द्वारा योजनाबद्ध रूप से दर्शाया गया है, इलेक्ट्रॉन की न्यूनतम गतिज ऊर्जा का आकलन कीजिए । (ii) उपर्युक्त नाभिक में सीमित न्यूट्रॉन की न्यूनतम गतिज ऊर्जा का आकलन कीजिए । व्याख्या कीजिए कि क्या एक इलेक्ट्रॉन को नाभिक के अंदर सीमित किया जा सकता है । 15 अंक

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Directive word: Prove

This question asks you to prove. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

Begin with the directive 'prove' for part (a), employing rigorous commutation algebra; allocate approximately 40% time to part (a) (20 marks) covering all four commutator identities systematically, 30% to part (b) (15 marks) for differentiation and maximization of probability density, and 30% to part (c) (15 marks) for particle-in-a-box energy calculations with proper unit conversions. Structure as: (a) state definitions of L±, L², Lz in position/momentum representation then derive each identity; (b) differentiate |ψ₀|², set to zero, verify maximum, compute numerical value; (c) apply E₁ = π²ℏ²/2mL² for both particles, compare with electron rest energy to demonstrate impossibility of electron confinement.

Key points expected

  • Part (a)(i)-(iv): Correct definition of angular momentum operators Lx, Ly, Lz in terms of position and momentum operators, and systematic application of canonical commutation relations [xi, pj] = iℏδij to prove all four identities
  • Part (b)(i): Differentiation of probability density P(x) = |ψ₀(x)|² with respect to x, setting dP/dx = 0 to find x = 0 as the only critical point, and verification via second derivative that this is a maximum
  • Part (b)(ii): Substitution of x = 0 into P(x) to obtain P_max = (mω/ℏπ)^(1/2), with proper handling of normalization constants
  • Part (c)(i): Application of ground state energy formula for infinite square well E₁ = π²ℏ²/2meL² with L = 10⁻¹⁵ m, yielding E₁ ≈ 150-200 MeV (order of magnitude correct)
  • Part (c)(ii): Calculation of neutron ground state energy E₁ ≈ 20-30 MeV using mn ≈ 2000 me, comparison with electron case, and physical explanation using Heisenberg uncertainty principle that electron confinement requires energy exceeding its rest mass (0.511 MeV), making confinement impossible
  • Explicit statement of ladder operator definitions L± = Lx ± iLy and their hermiticity properties in part (a)
  • Clear dimensional analysis and conversion to electron-volts in part (c) with recognition that ~150 MeV >> 0.511 MeV violates energy-momentum conservation for electrons

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies that [L², Lz] = 0 reflects simultaneous measurability of magnitude and z-component; recognizes L± as raising/lowering operators; correctly identifies Gaussian nature of harmonic oscillator ground state; understands that infinite well minimum energy arises from confinement (uncertainty principle); recognizes electron rest energy constraintStates correct final results but with muddled conceptual justification; confuses raising and lowering operators; identifies maximum at x=0 without clear reasoning; calculates energies but misses physical significance of electron mass-energy relationFundamental misconceptions about operator algebra (e.g., treating Lx, Ly as commuting); incorrect identification of maximum probability location; uses classical formulas for quantum energies; fails to recognize relativistic constraint on electron confinement
Derivation rigour25%12.5Step-by-step commutation with explicit [xi, pj] = iℏδij usage; proper expansion of L² = Lx² + Ly² + Lz²; systematic handling of operator products; complete differentiation chain for probability density; clear derivation of E_n = n²π²ℏ²/2mL² from Schrödinger equation boundary conditionsCorrect final expressions but skips intermediate steps (e.g., jumps from [Lx, Ly] = iℏLz to results without showing cross terms); minor algebraic errors that don't affect final structure; incomplete justification for boundary conditionsMissing critical steps (e.g., no use of canonical commutation relations); circular reasoning; mathematically invalid operations (e.g., treating operators as numbers); no derivation of energy formula, only stated
Diagram / FBD10%5Sketches: (a) vector diagram showing L, Lz, L± action on angular momentum cone; (b) Gaussian probability density |ψ₀|² vs x with maximum at origin marked; (c) infinite square well potential V(x) with ground state wavefunction ψ₁(x) = √(2/L) sin(πx/L) and energy level E₁ indicatedTwo of three required diagrams present but poorly labeled; or diagrams present without clear connection to derived quantities; schematic well diagram without wavefunctionNo diagrams despite visualizable content; or completely irrelevant diagrams; diagrams with wrong qualitative features (e.g., node at center for ground state)
Numerical accuracy25%12.5Part (b)(ii): P_max = (mω/πℏ)^(1/2) with correct exponent arithmetic; Part (c): E₁(electron) ≈ 150-200 MeV (using ℏc = 197 MeV·fm, L = 1 fm); E₁(neutron) ≈ 20-30 MeV; explicit comparison showing E_electron >> m_ec² = 0.511 MeV; correct order-of-magnitude estimates with proper significant figuresCorrect formulas but arithmetic errors in final values (factor of 2-4 errors); correct orders of magnitude but wrong units; missing factor of π² or 2 in energy formula; correct electron energy but incorrect neutron scaling (forgetting mass ratio)Orders of magnitude wrong (e.g., keV instead of MeV); incorrect formulas leading to nonsensical results; no numerical values despite question demands; confusion between ℏ and h in calculations
Physical interpretation20%10For (a): explains that commuting observables share eigenstates, enabling simultaneous measurement of L² and Lz; for (b): connects Gaussian to minimum uncertainty state; for (c): explains Heisenberg uncertainty ΔxΔp ≥ ℏ/2 → minimum kinetic energy from confinement; explicitly states electron confinement impossible because required energy (~150 MeV) exceeds rest mass energy, implying pair production; neutron confinement viable as E₁ << m_nc²States that electron cannot be confined without explaining energy-mass relation; mentions uncertainty principle without connecting to calculation; describes results without physical insightNo physical interpretation provided; claims electron can be confined; confuses kinetic and total energy; misapplies uncertainty principle (e.g., Δx = L instead of L/2 or similar)

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