Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
The Hamiltonian of a system of N particles in volume V at temperature T is
H = ∑ aiqi
2
2N
i=1
+ ∑ bipi
2
2N
i=1
where ai and bi are positive constants. The ensemble average of the Hamiltonian is
𝛼NkBT, where kB is the Boltzmann constant. The value of 𝛼 is ____ (in integer).
Official Answer Key:
2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.35)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Classical Mechanics** within *Section 2: Classical Mechanics* (1 Mark, NAT).
**2. Quantitative Numerical Calculation**:
- Identifying known experimental values and physical constants for Classical 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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