= Solution
<orthogonality> of angular modes makes the one-body term $\mathcal E a_1^\dagger a_1$, where
$$
\boxed{\mathcal E=\frac{\hbar^2}{2mR^2}}.
$$
The contact-interaction <integral> supplies $(U/4\pi)\delta_{l_1+l_2,l_3+l_4}$, expressing <angular momentum conservation>. Within modes zero and one, the four mixed permutations each reduce to $a_1^\dagger a_0^\dagger a_0a_1$. The two remaining terms have all four indices equal. Thus
$$
H=\mathcal E N_1+\frac U{4\pi}\left[N_0(N_0-1)+N_1(N_1-1)+4N_0N_1\right].
$$
This is the projected two-mode <Quantum Hamiltonian>; the full interaction could scatter into excluded modes, so the truncation is an assumed approximation rather than an exact invariant subspace of the complete ring problem.
Back to article page
