GATE PH 2026 • Q51 MSQ • 2 Marks

Section 7: Solid State Physics

Crystal Structure & Lattice Dynamics

Subtopic: Bragg's Law & X-ray Diffraction (Laue and Powder)

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:

  • (A) If B̂ is Hermitian, then  is unitary.
  • (B) If B̂ is anti-Hermitian, then  is unitary.
  • (C) If B̂ is Hermitian, then |Det(Â)| = 1.
  • (D) If B̂ is anti-Hermitian, then  is Hermitian.

Official Answer Key:

A;C;D

Step-by-Step Solution & Explanation

### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.51)** **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): `If B̂ is anti-Hermitian, then  is unitary.`**.

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