A three-dimensional classical Harmonic Oscillator is in thermal equilibrium with a heat reservoir at temperature T. Its average total energy would be :
MH SET - 2020, SET Exam - 2020,
Answer: D
The average total energy is 3 \(k_{B}T\).
The energy of a three-dimensional classical oscillator has six quadratic terms: three kinetic, \(p_{x}^{2}/2m\) and so on, and three potential, \(\frac{1}{2}kx^{2}\) and so on. By the equipartition theorem each quadratic term contributes \(\frac{1}{2}k_{B}T\):
\[ \langle E \rangle = 6 \times \frac{1}{2}k_{B}T = 3k_{B}T \]
\(\frac{3}{2}k_{B}T\) is the kinetic energy alone, as for a free particle. \(k_{B}T\) is a one-dimensional oscillator, and 6 \(k_{B}T\) wrongly gives each quadratic term a full \(k_{B}T\).
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