Q4
(a) A particle of rest mass m₀ has a kinetic energy K, show that its de Broglie wavelength is given by λ = hc/√[K(K+2m₀c²)] Hence calculate the wavelength of an electron of kinetic energy 2 MeV. What will be the value of λ if K<< m₀c² ? (15 marks) (b) Calculate the probability of finding a simple harmonic oscillator within the classical limits if the oscillator is in its normal state. Also show that if the oscillator is in its normal state, then the probability of finding the particle outside the classical limits is approximately 16%. (15 marks) (c) Describe normal and anomalous Zeeman effect. Explain how it lifts the degeneracy in hydrogen atom. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) विराम-द्रव्यमान m₀ के एक कण की गतिज ऊर्जा K है, दर्शाइए कि इसकी डी. ब्रोगली तरंगदैर्घ्य निम्नलिखित द्वारा निर्धारित है : λ = hc/√[K(K+2m₀c²)] अतः 2 MeV गतिज ऊर्जा के एक इलेक्ट्रॉन के तरंगदैर्घ्य की गणना कीजिए । यदि K<< m₀c² हो तो λ का मान क्या होगा ? (15 अंक) (b) चिरप्रतिबंधित सीमाओं के अंदर एक सरल आवर्ती दोलक की प्रायिकता की गणना कीजिए यदि दोलक अपनी सामान्य अवस्था में है । यह भी दर्शाइए कि यदि दोलक अपनी सामान्य अवस्था में है तो कण के चिरप्रतिबंधित सीमा से बाहर निकलने की प्रायिकता लगभग 16% (प्रतिशत) है । (15 अंक) (c) सामान्य और असंगत ज़ीमान प्रभाव का वर्णन कीजिए । समझाइए कि यह हाइड्रोजन परमाणु में अपभ्रष्टता को कैसे उठा देता है । (20 अंक)
Directive word: Derive
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How this answer will be evaluated
Approach
Begin with the relativistic energy-momentum relation to derive the de Broglie wavelength formula in part (a), then transition to quantum mechanical probability calculations for the harmonic oscillator in part (b), and conclude with a systematic exposition of Zeeman effects and degeneracy lifting in part (c). Allocate approximately 30% effort to part (a) due to its dual derivation-cum-calculation demand, 30% to part (b) for its integration-heavy probability analysis, and 40% to part (c) given its higher mark weightage and conceptual depth requiring clear energy level diagrams.
Key points expected
- Part (a): Derivation using E² = p²c² + m₀²c⁴ with E = K + m₀c², leading to λ = hc/√[K(K+2m₀c²)]; calculation of electron wavelength at 2 MeV (≈ 0.0056 Å or similar); non-relativistic limit reduction to λ = h/√(2m₀K)
- Part (b): Classical turning points for ground state SHO at x = ±√(ℏ/mω); probability integral P = ∫|ψ₀|²dx between -x₀ to +x₀ using Gaussian integral; demonstration that P ≈ 0.84 inside, hence ~16% outside classical limits
- Part (c): Normal Zeeman effect (spin singlet, orbital only, triplet splitting) vs anomalous Zeeman effect (spin-orbit coupling, Paschen-Back transition); role of g-factor (Landé g-factor for anomalous case); lifting of l-degeneracy in hydrogen via m_l quantum number and selection rules Δm = 0, ±1
- Clear distinction between strong and weak field regimes for Zeeman effects, with mention of Indian physicist S. Panck or relevant experimental context where applicable
- Proper handling of units throughout (MeV, Å, natural constants) and dimensional consistency checks
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 10 | Demonstrates flawless command of relativistic kinematics in (a), quantum probability interpretation in (b), and atomic spectroscopy including LS coupling in (c); correctly identifies when non-relativistic approximation fails and distinguishes normal/anomalous regimes by spin multiplicity | Shows adequate understanding of core concepts but confuses relativistic vs non-relativistic regimes, or conflates normal/anomalous Zeeman effects, or misidentifies classical turning points | Fundamental misconceptions: treats kinetic energy as ½mv² in relativistic regime, or defines classical limits incorrectly, or describes Zeeman effect without mentioning magnetic quantum number |
| Derivation rigour | 25% | 12.5 | Presents complete, step-by-step derivations: energy-momentum relation → momentum expression → wavelength in (a); explicit ψ₀ normalization and error function integral setup in (b); Hamiltonian perturbation derivation with explicit g-factor formula in (c); all algebraic steps justified | Derivations present but with skipped steps or unmotivated substitutions; correct final results but unclear logical flow; partial integration setup in (b) without completing to numerical value | Missing derivations entirely or logically circular; jumps from given to result without intermediate steps; incorrect algebraic manipulation leading to wrong functional forms |
| Diagram / FBD | 15% | 7.5 | Includes clear, labeled diagrams: energy-momentum hyperbola sketch for relativistic relation in (a); probability density |ψ₀|² plot with classical turning points marked in (b); detailed energy level diagram showing l=1 splitting into m_l = -1,0,+1 states with/without spin for normal vs anomalous Zeeman in (c) | Basic diagrams present but poorly labeled or missing key features; generic Zeeman diagram without distinguishing normal/anomalous cases; no probability density visualization | No diagrams despite clear need for visualization, or completely irrelevant sketches; fails to illustrate degeneracy lifting graphically |
| Numerical accuracy | 20% | 10 | Precise calculation in (a): λ = hc/√[2×(2+0.511)×(2)²] MeV² → correct value in Å or nm with 2-3 significant figures; exact probability value ~0.8427 in (b) leading to ~15.73% outside; consistent use of ℏc = 197.3 MeV·fm or equivalent | Correct order of magnitude but arithmetic errors; approximate probability ~0.84 without precise calculation; unit conversion errors (eV vs MeV) caught and partially corrected | Order-of-magnitude errors; missing numerical evaluation entirely; incorrect substitution of constants; probability calculation absent or completely wrong |
| Physical interpretation | 20% | 10 | Insightful commentary: explains why relativistic formula matters for MeV electrons (comparable to rest mass); interprets quantum tunneling/classically forbidden regions in (b); connects Zeeman splitting to spectroscopic observation (doublet/triplet), mentions applications like astrophysical magnetic field measurement or precision spectroscopy | Basic interpretation present but superficial; states results without explaining significance; limited connection to experimental observables | No physical interpretation provided; purely mathematical treatment; fails to explain what degeneracy lifting means for observable spectra |
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