GATE PH 2026 • Q51
MCQ • 2 Marks
Section 9: Nuclear and Particle Physics
Nuclear & Particle Physics
Subtopic: Nuclear Models, Radioactivity & Elementary Particles
Question Statement:
Consider two operators  and B̂ which are related as  = exp(i𝜃B̂). If 𝜃 is a non-zero
real number, which of the following statements is/are true?
Options:
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(A) If B̂ is Hermitian, then  is unitary.
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(B) If B̂ is anti-Hermitian, then  is unitary.
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(C) If B̂ is Hermitian, then |Det(Â)| = 1.
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(D) If B̂ is anti-Hermitian, then  is Hermitian.
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.51)**
**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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