GATE PH 2026 • Q50
MSQ • 2 Marks
Section 7: Solid State Physics
Crystal Structure & Lattice Dynamics
Subtopic: Bragg's Law & X-ray Diffraction (Laue and Powder)
Question Statement:
Which of the following operators is/are self-adjoint?
Options:
-
(A) x2 d2
dx2 + 3x d
dx + x2
-
(B) (1 - x2) d2
dx2 - 2x d
dx + 3x
-
(C) (3x - 4x3) d2
dx2 + (3 - 12x2) d
dx + 12
-
(D) x d2
dx2 + x2 d
dx + 5x
3
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.50)**
**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): `(1 - x2) d2
dx2 - 2x d
dx + 3x`**.
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