Consider operators Â, B̂, and Ĉ for three observables of a quantum system satisfying
[Â, B̂] = 0, [B̂, Ĉ] = 0, and [Â, Ĉ] ≠ 0, with uncertainties ΔA, ΔB, ΔC, respectively. From
the options given below, which is/are implied by the commutation relations among
Â, B̂, and Ĉ ?
Options:
(A) ΔA ΔB > 0
(B) ΔA ΔC > 0
(C) Â, B̂ can be simultaneously diagonalized.
(D) Â, B̂, Ĉ can be simultaneously diagonalized.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.52)**
**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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