Physics 2022 Paper II 50 marks Derive

Q2

(a) Set up the Schrodinger's wave equation for one dimensional potential barrier and obtain the probability of tunneling. 20 marks (b) Show that $E_n = <V>$ in the stationary states of the hydrogen atom. 15 marks (c) (i) Show that for a given principal quantum number $n$, there are $n^2$ possible states of the atom. (ii) An atomic state is denoted by $^4D_{5/2}$. Find the values of $L$, $S$ and $J$. For this state, what should be the minimum number of electrons involved ? Suggest a possible electronic configuration. 7+8=15 marks

हिंदी में प्रश्न पढ़ें

(a) एक आयामी विभव प्राचीर के लिए श्रोडिंगर का तरंग समीकरण स्थापित कीजिए और इसकी सुरंगन (टनलिंग) की संभावना ज्ञात कीजिये । 20 अंक (b) दिखाइए कि हाइड्रोजन परमाणु के स्थायी अवस्थाओं में $E_n = <V>$ होता है । 15 अंक (c) (i) दिखाइए कि एक दिये गये मुख्य क्वांटम संख्या $n$ के लिए परमाणु की संभावित अवस्थायें $n^2$ होती हैं । (ii) एक परमाणु अवस्था को $^4D_{5/2}$ द्वारा निर्दिष्ट किया जाता है, तो : $L$, $S$ और $J$ का मान ज्ञात कीजिए । इस अवस्था के लिए शामिल इलेक्ट्रॉनों की न्यूनतम संख्या कितनी होनी चाहिए ? एक संभावित इलेक्ट्रॉनिक संरचना का सुझाव दीजिए । 7+8=15 अंक

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Directive word: Derive

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Approach

Derive the tunneling probability for a 1D potential barrier in part (a) using boundary condition matching; prove the virial theorem relation for hydrogen in (b); enumerate n² degeneracy with proper quantum number counting for (c)(i); and decode the term symbol ^4D_{5/2} to extract L, S, J and suggest a minimal electron configuration for (c)(ii). Allocate approximately 40% time to (a), 30% to (b), and 30% combined to both parts of (c), ensuring each sub-part receives proportional attention to its marks.

Key points expected

  • Part (a): Set up time-independent Schrödinger equation for regions I (x<0), II (0<x<a), and III (x>a) with appropriate wave functions; apply continuity conditions for ψ and dψ/dx at x=0 and x=a; obtain transmission coefficient T = |F/A|² = [1 + (k₁²+k₂²)²sinh²(κa)/(4k₁²k₂²)]⁻¹ ≈ 16E(V₀-E)/V₀² exp(-2κa) for E<V₀
  • Part (b): Apply virial theorem 2⟨T⟩ = ⟨r·∇V⟩ for Coulomb potential V∝1/r; show ⟨T⟩ = -Eₙ/2 and ⟨V⟩ = 2Eₙ; conclude Eₙ = ⟨V⟩/2 + ⟨T⟩ = ⟨V⟩ - ⟨V⟩/2 = ⟨V⟩/2 + ⟨V⟩/2 verification or direct proof using hydrogenic wave functions and expectation value ⟨r⁻¹⟩ = 1/(n²a₀)
  • Part (c)(i): Count states using quantum numbers n, l, m_l: for given n, l ranges 0 to n-1; each l has (2l+1) m_l values; sum Σ(2l+1) from l=0 to n-1 equals n²; alternatively include spin degeneracy factor of 2 to get 2n² for total states
  • Part (c)(ii): Decode term symbol ^4D_{5/2}: 2S+1=4 gives S=3/2; D means L=2; J=5/2; minimum electrons = 3 (for S=3/2 requires at least 3 unpaired electrons); possible configuration: [Ar]3d³4s² (vanadium) or [Ar]3d⁷4s² (cobalt) or 2p²3p¹ excited state
  • Physical significance: Connect tunneling to α-decay (Gamow factor), STM microscopy, and quantum devices; relate hydrogen result to stability of atomic orbits; connect term symbols to atomic spectroscopy and Hund's rules applications

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies evanescent wave nature in barrier region for (a); accurately states virial theorem for inverse-square law forces in (b); properly counts degeneracy including all quantum numbers for (c)(i); correctly interprets term symbol multiplicity and identifies minimum electron count as 3 for (c)(ii)Sets up basic equations but confuses wave numbers in different regions; states virial theorem without proper justification; counts states but misses spin or orbital degeneracy factors; decodes L and J correctly but errs on S value or minimum electron countUses wrong potential form or confuses bound/unbound states; applies virial theorem incorrectly to harmonic oscillator; simply states n² without derivation; fails to decode term symbol or gives wrong values for all quantum numbers
Derivation rigour25%12.5Complete step-by-step derivation with all four boundary conditions explicitly applied for (a); elegant proof using either virial theorem operator or direct integration for (b); clear algebraic summation Σ(2l+1)=n² with explicit steps for (c)(i); logical deduction of minimum electrons from spin coupling rules for (c)(ii)Sets up boundary conditions but skips algebraic simplification; states key results without showing intermediate steps; gives final summation formula without showing l-by-l breakdown; gives correct quantum numbers but weak justification for minimum electron countMissing critical steps like boundary condition matching; no derivation, only final formulas; incorrect summation or no derivation; no logical connection between term symbol and electron configuration
Diagram / FBD15%7.5Clear potential energy diagram showing V₀, regions I-III, incident/reflected/transmitted waves with proper labeling for (a); energy level diagram showing Eₙ, ⟨T⟩, ⟨V⟩ relationship for (b); quantum number hierarchy diagram or table for (c)(i); term symbol coupling diagram showing L-S coupling and vector addition for (c)(ii)Basic potential barrier sketch without wave function annotations; mentions diagram but doesn't draw; simple table of quantum numbers without visual hierarchy; sketches orbital diagram but unclear on coupling schemeNo diagrams despite clear need; diagrams with wrong potential shape or missing regions; no visual aid for degeneracy counting; no attempt to illustrate term symbol meaning
Numerical accuracy20%10Correct final tunneling probability formula with proper exponential dependence; accurate numerical coefficients in simplified form; precise statement of expectation values; exact values L=2, S=3/2, J=5/2 with correct arithmetic for 2S+1=4; valid electron count of 3Correct functional form but wrong prefactor; correct final relation but missing factor of 2 somewhere; correct n² result but arithmetic errors in summation; correct L and J but S=1 instead of 3/2Wrong formula (e.g., using sin instead of sinh); order-of-magnitude errors; algebraic mistakes in summation; completely wrong numerical values for quantum numbers
Physical interpretation20%10Explains tunneling as wave penetration with exponential decay, cites α-decay (Geiger-Nuttall law) or STM applications; clarifies that Eₙ=⟨V⟩ reflects virial theorem balance of kinetic and potential energies; explains n² degeneracy removal by external fields (Stark, Zeeman); connects ^4D_{5/2} to Hund's rules and ground state determination for transition metals like V or CoBrief mention of tunneling applications without elaboration; states virial theorem without physical insight; mentions degeneracy without physical consequence; identifies configuration but doesn't discuss Hund's rulesNo physical interpretation of tunneling; treats Eₙ=⟨V⟩ as mere algebraic coincidence; no discussion of why degeneracy counting matters; no connection to real atoms or spectroscopic applications

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