= Planar rigid rotor
{title2=$E_m=\hbar^2m^2/(2I)$}
= Planar rotor
{synonym}
= Two-dimensional rigid rotor
{synonym}
A <rigid rotor> restricted to one angular coordinate $\phi$. Single-valued spatial states $e^{im\phi}$ have $m\in\mathbb Z$ and <energy> $E_m=\hbar^2m^2/(2I)$, with <moment of inertia> $I$. The zero-angular-momentum level has <state degeneracy> one; every nonzero $|m|$ has two directions. For labelled molecular ends its high-temperature <rotational partition function> is $\sqrt{\pi T/\theta_{\rm rot}}$, where $\theta_{\rm rot}=\hbar^2/(2Ik_{\rm B})$. Equivalent orientations and <nuclear spin statistics> require a further molecular counting convention. One quadratic <angular momentum> gives rotational <heat capacity> $k_{\rm B}/2$ per molecule, rather than $k_{\rm B}$ for an unrestricted <linear rigid rotor>.
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