= Solution
For a <particle in a ring>, the arc-length coordinate is $s=R\phi$. The tangential <kinetic energy> is
$$
H=\frac{p_s^2}{2m}=\frac{L_z^2}{2mR^2}.
$$
The constraint removes radial motion; the constant <potential energy> along the ring can be chosen as zero. The tangential <momentum operator> is $\hat p_s=-i\hbar\partial_s$, or equivalently the <angular momentum operator> is $\hat L_z=-i\hbar\partial_\phi$. Thus
$$
\boxed{\hat H=-\frac{\hbar^2}{2mR^2}\frac{\partial^2}{\partial\phi^2}.}
$$
Its physical domain has periodic <boundary conditions>, $\psi(\phi+2\pi)=\psi(\phi)$ and $\psi'(\phi+2\pi)=\psi'(\phi)$. These make the <Hamiltonian> <self-adjoint> and reflect that $\phi$ and $\phi+2\pi$ describe the same point. No magnetic flux or additional angle-dependent <potential energy> is present.
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