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GATE 2023 PH – Question 38

Thermodynamics and Statistical Mechanics · partition function, free energy, calculation of thermodynamic quantities · 2 marks · Multiple choice

A harmonic oscillator has $\omega=2k_BT/\hbar$ in thermal equilibrium at temperature T. Which is its partition function?

  1. $e/(e^2-1)$
  2. $e/(e^2+1)$
  3. $e/(e-1)$
  4. $e/(e+1)$

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Correct answer: (A) $e/(e^2-1)$

Explanation

The partition function of a quantum harmonic oscillator (energies $E_n=(n+\tfrac12)\hbar\omega$) is
$$Z=\sum_{n=0}^\infty e^{-\beta E_n}=\frac{e^{-\beta\hbar\omega/2}}{1-e^{-\beta\hbar\omega}},\qquad\beta=\frac1{k_BT}.$$

**Numbers.** With $\omega=2k_BT/\hbar$, $\beta\hbar\omega=2$:
$$Z=\frac{e^{-1}}{1-e^{-2}}=\frac{e^{-1}\cdot e^{2}}{e^{2}-1}=\frac{e}{e^2-1}\quad(\text{option A}).$$