Subtopic: Bragg's Law & X-ray Diffraction (Laue and Powder)
Question Statement:
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;C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.52)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 7: Solid State Physics* | **Topic**: *Crystal Structure & Lattice Dynamics* (`Bragg's Law & X-ray Diffraction (Laue and Powder)`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Crystal Structure & Lattice Dynamics** (`Bragg's Law & X-ray Diffraction (Laue and Powder)`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `ΔA ΔC > 0`**.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.