Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 33B / c / Solution

Write the equally spaced one-particle spectrum asThe two-particle level labelled by $N$ has energyand receives contributions from orbital pairs $(n_1,n_2)$ with $n_1+n_2=N$.

For every unequal unordered pair $n_1<n_2$, there is one symmetric and one antisymmetric spatial combination. The symmetric one combines with the six-dimensional symmetric spin space; the antisymmetric one combines with the three-dimensional antisymmetric spin space. Thus each unequal pair contributes nine states.

If $N$ is odd, all pairs are unequal and there are $(N+1)/2$ unordered pairs. Therefore

If $N$ is even, there are $N/2$ unequal pairs and the additional equal pair $n_1=n_2=N/2$. The latter has only a symmetric spatial state and therefore contributes six spin states. HenceThis is the degeneracy of two identical spin-one bosons with equally spaced orbital levels.

Solved by gpt-5.6-sol high.