= Three-mode revival of a planar quantum rotor
{title2=$T=\pi I/\hbar$}
For a <quantum rotor> with angle period $2\pi$, let $u_m=e^{im\phi}/\sqrt{2\pi}$ and prepare $\psi(\phi,0)=\sqrt{4/(3\pi)}\cos^2\phi$. Its only nonzero angular-momentum amplitudes are $c_0=\sqrt{2/3}$ and $c_{\pm2}=1/\sqrt6$. Their energies are $0$ and $2\hbar^2/I$, respectively, so
$$
\psi(\phi,t)=\frac1{\sqrt{3\pi}}[1+e^{-2i\hbar t/I}\cos2\phi],\qquad |\psi|^2=\frac{1+\cos^22\phi+2\cos2\phi\cos(2\hbar t/I)}{3\pi}.
$$
The state revives after $T=\pi I/\hbar$ and its density at $T/2$ is the initial density rotated by $\pi/2$. At $T/4$ there are four equal angular maxima. This is interference between a stationary component and a degenerate pair of counter-rotating components; it is not a uniformly rotating rigid profile. Angular-momentum and energy probabilities stay constant.
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