GATE 2024 PH – Question 35
A system has $H=\sum_{i=1}^{2N}a_iq_i^2+\sum_{i=1}^{2N}b_ip_i^2$, with positive constants. Its ensemble average is $\alpha Nk_BT$. The integer $\alpha$ is _____
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Correct answer: 2
Explanation
**Equipartition theorem.** Each term in the Hamiltonian that is **quadratic** in a coordinate or momentum contributes $\tfrac12k_BT$ to the ensemble-average energy.
The Hamiltonian
$$H=\sum_{i=1}^{2N}a_iq_i^2+\sum_{i=1}^{2N}b_ip_i^2$$
has $2N$ quadratic position terms and $2N$ quadratic momentum terms, that is $4N$ quadratic terms in all.
$$\langle H\rangle=4N\times\tfrac12k_BT=2Nk_BT=\alpha Nk_BT\;\Rightarrow\;\alpha=\mathbf{2}.$$