Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
At a certain temperature T, the average speed of nitrogen molecules in air is found to be 400 m/s.
The most probable and the root mean square speeds of the molecules are, respectively,
(A) 355 m/s, 434 m/s (B) 820 m/s, 917 m/s
(C) 152 m/s, 301 m/s (D) 422 m/s, 600 m/s
Common Data Questions
Common Data for Questions 48 and 49:
The wavefunction of a particle moving in free space is given by,
Official Answer Key:
2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2012 Q.47)**
**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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