Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
The probability current density for the real part of the wavefunction is
(A) 1 (B)
k
m
(C)
2
k
m
(D) 0
Common Data for Questions 50 and 51:
The dispersion relation for a one dimensional monatomic crystal with lattice spacing a , which interacts via
nearest neighbour harmonic potential is given by
,
where A is a constant of appropriate unit.
Official Answer Key:
2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2012 Q.49)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (2 Marks, NAT).
**2. Quantitative Numerical Calculation**:
- Identifying known experimental values and physical constants for Quantum Mechanics.
- Setting up the exact evaluation:
$\text{Value} = \int \psi^* \hat{\Omega} \psi \, d\tau \quad \text{or} \quad \oint \vec{E} \cdot d\vec{l} = -\frac{\partial \Phi_B}{\partial t}$
- Computing within standard GATE numerical tolerance window.
**3. Final Answer**:
Official numerical answer is **2.5**.
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