A meson consists of two quarks, attracted by a linear potential energy

$$V=\alpha x$$

where $x$ is the separation between the quarks and $\alpha$ is a constant. Treating the quarks classically, compute the vibrational partition function that arises from the separation of quarks. What is the average separation of the quarks at temperature $T$ ?

Consider an ideal gas of these mesons that have the orientation of the quarks fixed so the mesons do not rotate. Compute the total partition function of the gas. What is its heat capacity $C_{V}$ ?

[Note: $\int_{-\infty}^{+\infty} d x e^{-a x^{2}}=\sqrt{\pi / a} .$ ]