GATE PH 2024 • Q47 MCQ • 2 Marks

Section 9: Nuclear and Particle Physics

Nuclear & Particle Physics

Subtopic: Nuclear Models, Radioactivity & Elementary Particles

Question Statement:

A particle of mass m is moving in the potential V(x) = {V0 + 1 2 m𝜔0 2x2, x > 0, ∞ , x ≤ 0, Figures P, Q, R and S show different combinations of the values of 𝜔0 and V0. Ej (P), Ej (Q), Ej (R) and Ej (S) with j = 0, 1, 2, …, are the eigen-energies of the j-th level for the potentials shown in figures P, Q, R and S, respectively. W hich of the statement is/are true?

Options:

  • (A) E0 (P) = E0 (Q)
  • (B) E0 (Q) = E0 (S)
  • (C) E0 (P) = E1 (R)
  • (D) E0 (R) ≠ E0 (Q)

Official Answer Key:

B

Step-by-Step Solution & Explanation

### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.47)** **1. Theoretical Formulation & Physics Principles**: This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (2 Marks, MCQ). **2. Analytical Derivation & Formalism**: - Formulating the fundamental physics relations governing Nuclear & Particle Physics: - Applying canonical quantum/electrodynamic equations: $\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$ - Substituting boundary parameters and simplifying algebraically. - The resulting expression uniquely corresponds to Option **(B)**. **3. Final Answer**: Correct Option is **(B)**.

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