Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 5 / a / Solution

orthogonality of angular modes makes the one-body term $\mathcal{E}{a}_1^{†}{a}_1$, whereThe 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 $a_1^{†}{a}_0^{†}{a}_0{a}_1$. The two remaining terms have all four indices equal. ThusThis 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.