= Solution
With angular-coordinate measure $d\phi$, the normalized one-particle orbitals are $u_l(\phi)=e^{il\phi}/\sqrt{2\pi}$. Single-valuedness requires integer $l$. Their products obey
$$
\int_0^{2\pi}\!\!\int_0^{2\pi}|\psi|^2\,d\phi_1d\phi_2=1,\qquad
\boxed{E_{l_1l_2}=\epsilon_R(l_1^2+l_2^2),\quad
\epsilon_R=\frac{\hbar^2}{2mR^2}}.
$$
If arc-length measure is used instead, each one-particle <wavefunction normalization> includes $R^{-1/2}$; the physical <energy> is unchanged. For the following parts impose antisymmetry on the full spatial-and-spin state of the two <fermions>.
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