 Solution

ID: 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

E a_{1}^{†} a_{1}

, where

E = \frac{ℏ ^{2}}{2 m R ^{2}} .

The contact-interaction integral supplies

(U /4 π) δ_{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. Thus

H = E N_{1} + \frac{U}{4 π} [N_{0} (N_{0} - 1) + N_{1} (N_{1} - 1) + 4 N_{0} N_{1}] .

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.

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