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For a particle in a ring, the arc-length coordinate is $s=R\varphi$. The tangential kinetic energy isThe constraint removes radial motion; the constant potential energy along the ring can be chosen as zero. The tangential momentum operator is ${\hat{p}}_s=-i\hslash {\partial }_s$, or equivalently the angular momentum operator is ${\hat{L}}_z=-i\hslash {\partial }_{\varphi }$. ThusIts physical domain has periodic boundary conditions, $\psi (\varphi +2\pi )=\psi (\varphi )$ and ${\psi }^{\prime }(\varphi +2\pi )={\psi }^{\prime }(\varphi )$. These make the Hamiltonian self-adjoint and reflect that $\varphi$ and $\varphi +2\pi$ describe the same point. No magnetic flux or additional angle-dependent potential energy is present.