Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
Consider two hypothetical nuclei X1 and X2 undergoing 𝛽 decay, resulting in nuclei
Y1 and Y2, respectively. The decay scheme and the corresponding JP values of the
nuclei are given in the figure. Which of the following option(s) is/are correct? (J is
the total angular momentum and P is parity)
Options:
(A) X1 → Y1 is Fermi transition and X2 → Y2 is Fermi transition
(B) X1 → Y1 is Fermi transition and X2 → Y2 is Gamow-Teller transition
(C) X1 → Y1 is Gamow-Teller transition and X2 → Y2 is Fermi transition
(D) X1 → Y1 is Gamow-Teller transition and X2 → Y2 is Gamow-Teller transition
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.48)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Quantum Mechanics:
- 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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