Question

A simple harmonic oscillator with an angular frequency $\omega$ is in thermal equilibrium with a reservoir at absolute temperature $T$, with $\omega = \dfrac{2\pi k_B T}{h}$. Which one of the following is the partition function of the system?

• A. \(\frac{e}{e^2 - 1}\)
• B. \(\frac{e}{e^2 + 1}\)
• C. \(\frac{e}{e - 1}\)
• D. \(\frac{e}{e + 1}\)

Hint:

For a quantum harmonic oscillator in thermal equilibrium, the partition function is derived based on the Boltzmann distribution and the energy levels of the system.

Answer 1

The Correct Option is A

- The partition function \( Z \) for a quantum harmonic oscillator is given by: \[ Z = \frac{e^{\frac{\hbar \omega}{k_B T}}}{1 - e^{-\frac{\hbar \omega}{k_B T}}} \] - Given the expression for \(\omega\), we substitute and simplify the partition function expression to match the available options.