Physics 2023 Paper II 50 marks Compulsory Calculate

Q1

(a) Calculate the zero point energy for a particle in an infinite potential well for the following cases : (i) a 100 g ball confined on a 5 m long line. (ii) an oxygen atom confined to a 2×10⁻¹⁰ m lattice. (iii) an electron confined to a 10⁻¹⁰ m atom. Why zero point energy is not important for macroscopic objects ? Comment. 10 marks (b) Consider a particle of mass m and charge q moving under the influence of a one dimensional harmonic oscillator potential. Assume it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = p²/2m + ½mω²X² - qEX. Obtain the energy expression and the wave function of the nth excited state of the particle. 10 marks (c) A particle of mass m is in a spherically symmetric attractive potential of radius a. Find the minimum depth of the potential needed to have two bound states of zero angular momentum. 10 marks (d) A beam of hydrogen atoms emitted from an oven at 400 k is sent through a Stern-Gerlach experiment having magnet of length 1 m and a gradient field of 10 tesla/m. Calculate the transverse deflection of an atom at the point where the beam leaves the magnet. 10 marks (e) If an atom is placed in a magnetic field of strength 0·1 weber/m², then calculate the rate of precession and torque on an electron with l = 3 in the atom. Given that the magnetic moment of the electron makes an angle of 30° with the magnetic field. 10 marks

हिंदी में प्रश्न पढ़ें

(a) एक कण को अनंत विभव कूप में रखने पर निम्न स्थितियों के लिए शून्य बिंदु ऊर्जा की गणना करें : (i) एक 100 g की गेंद जो 5 m लंबी रेखा पर प्रतिबंधित है । (ii) एक ऑक्सीजन परमाणु जो 2×10⁻¹⁰ m जालक पर प्रतिबंधित है । (iii) एक इलेक्ट्रॉन जो 10⁻¹⁰ m परमाणु में प्रतिबंधित है । स्थूल वस्तुओं के लिए शून्य बिंदु ऊर्जा का महत्व क्यों नहीं है ? टिप्पणी करें । 10 अंक (b) द्रव्यमान m और आवेश q का एक कण एक एकविमीय आवर्ती दोलक विभव के प्रभाव के अधीन गतिशील है । मान लीजिये कि इसे एक स्थिर विद्युत क्षेत्र E में रखा गया है । इसलिये इस कण का हैमिल्टोनियन H = p²/2m + ½mω²X² - qEX द्वारा प्रदत है । कण की nth उत्तेजित अवस्था के लिये ऊर्जा व्यंजक और तरंग फलन प्राप्त कीजिये । 10 अंक (c) द्रव्यमान m का एक कण अर्धव्यास a के गोलीय सममित आकर्षक विभव में है । शून्य कोणीय संवेग की दो परिबद्ध अवस्थाओं के लिये आवश्यक विभव की न्यूनतम गहराई ज्ञात कीजिये । 10 अंक (d) तापमान 400 k पर एक अवन से उत्सर्जित हाइड्रोजन परमाणुओं का एक पुंज स्टर्न-गर्लैक प्रयोग में, जिसके चुंबक की लंबाई 1 m व चुंबकीय प्रवणता क्षेत्र 10 टेस्ला/मीटर है, भेजा जाता है । उस बिंदु पर जहाँ पुंज चुंबक को छोड़ता है, अनुप्रस्थ विषेषण की गणना कीजिये । 10 अंक (e) यदि एक परमाणु को 0·1 weber/m² तीव्रता के चुंबकीय क्षेत्र में रखा है, तब एक इलेक्ट्रॉन जो परमाणु में l = 3 अवस्था में है की पुरस्सरण की दर एवं बल आघूर्ण की गणना कीजिये। दिया गया है कि इलेक्ट्रॉन का चुंबकीय आघूर्ण चुंबकीय क्षेत्र से 30° का कोण बनाता है। 10 अंक

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Directive word: Calculate

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How this answer will be evaluated

Approach

Calculate requires systematic numerical working with proper physical reasoning. Structure: (a) Apply E₁ = π²ℏ²/(2mL²) for all three cases, showing orders of magnitude comparison; (b) Complete the square for the shifted harmonic oscillator, finding new equilibrium and energy levels; (c) Solve radial Schrödinger equation for l=0 with boundary conditions at r=a; (d) Use force method or quantum expectation for Stern-Gerlach splitting; (e) Apply Larmor precession formula τ = μ×B. Allocate ~20% time each to (a), (b), (c), (d), (e) as all carry equal marks.

Key points expected

  • (a) Zero-point energy calculation: E₁ = π²ℏ²/(2mL²) for 100g ball (~10⁻⁶⁷ J), oxygen atom (~10⁻²¹ J), electron (~10⁻¹⁸ J); explicit comparison showing macroscopic E₁ is negligible vs thermal energy kT ~ 10⁻²¹ J at 300K
  • (b) Shifted harmonic oscillator: complete square to get H = p²/2m + ½mω²(X - qE/mω²)² - q²E²/2mω²; energy levels Eₙ = (n+½)ℏω - q²E²/2mω²; wave functions ψₙ(x) = φₙ(x - qE/mω²) where φₙ are standard HO eigenfunctions
  • (c) Spherical well: radial equation for l=0 with u(r)=rR(r); boundary conditions u(0)=0, continuity of u and u' at r=a; transcendental equation for bound states; condition for two bound states requires at least one node, giving minimum depth V₀ ≥ π²ℏ²/(8ma²)
  • (d) Stern-Gerlach: force F = μ_z(dB/dz) = ±(eℏ/2m_e)(dB/dz); classical trajectory with transverse acceleration; deflection δz = (F/m_H)(L/v)²/2 where v = √(3kT/m_H); numerical value ~0.3-0.5 mm
  • (e) Larmor precession: ω_L = g_lμ_BB/ℏ = μ_BB/ℏ for orbital moment (g_l=1); torque |τ| = μBsinθ = lμ_B·B·sin30°; precession rate ~10⁹ rad/s, torque ~10⁻²³ Nm

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies quantum confinement in (a), displaced oscillator in (b), bound state condition in (c), spin-space separation in (d), and orbital magnetic moment vs spin distinction in (e); no confusion between infinite and finite wellsCorrect formulas but minor conceptual errors like treating (c) as infinite well or confusing spin and orbital magnetic moment in (e)Major conceptual errors: classical treatment for (a)-(c), missing quantum nature entirely, or using wrong magnetic moment in (d)-(e)
Derivation rigour20%10Complete derivations: explicit Schrödinger equation setup, boundary condition application, algebraic steps for completing square in (b), transcendental equation derivation in (c), trajectory integration in (d)Correct final formulas with gaps in derivation steps; missing explicit boundary condition justification in (c) or integration limits in (d)Jumps to final answers without derivation; no justification for wave function forms or energy level shifts
Diagram / FBD10%5Clear potential well diagrams for (a) and (c) showing wave functions; displaced parabola sketch for (b); Stern-Gerlach apparatus schematic with force direction in (d); precession cone diagram for (e)At least 2-3 relevant diagrams with correct qualitative features but missing labels or quantitative detailsNo diagrams or completely irrelevant sketches; missing all visual representations of potentials and wave functions
Numerical accuracy30%15All five numerical answers correct with proper SI units and significant figures: (a) three values with correct orders 10⁻⁶⁷J, 10⁻²¹J, 10⁻¹⁸J; (b) energy shift magnitude; (c) V₀ in terms of given parameters; (d) deflection ~0.3-0.5 mm; (e) ω ~10⁹ rad/s, τ ~10⁻²³ NmCorrect order of magnitude for most parts with 1-2 calculation errors or unit conversion mistakes; correct formulas with substitution errorsWrong orders of magnitude (especially missing ℏ² vs ℏ); incorrect constants; no numerical evaluation despite calculable quantities
Physical interpretation20%10Explicit comparison of zero-point energy to thermal energy in (a) explaining classical limit; physical meaning of displaced equilibrium in (b); interpretation of bound state number vs well depth in (c); quantum-classical correspondence in Stern-Gerlach (d); torque-precession relationship visualization in (e)Some physical insight but limited connection between mathematical results and observable phenomena; generic statements about quantum mechanicsPurely mathematical treatment with no physical interpretation; no explanation of why results matter or how they connect to experiments

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