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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 33C / c / Solution

For

T_{ij} = a_{i}^{†} a_{j}

, expand the commutator and commute one annihilation operator past one creation operator:

[T_{ij}, T_{k l}] = a_{i}^{†} a_{j} a_{k}^{†} a_{l} - a_{k}^{†} a_{l} a_{i}^{†} a_{j} = δ_{jk} a_{i}^{†} a_{l} - δ_{i l} a_{k}^{†} a_{j} .

Hence the oscillator bilinear commutator is

[T_{ij}, T_{k l}] = δ_{jk} T_{i l} - δ_{i l} T_{kj} .

Since

H = T_{xx} + T_{yy} + 1,

we obtain

[T_{ij}, H] = \sum_{k} (δ_{jk} T_{ik} - δ_{ik} T_{kj}) = 0.

Every

T_{ij}

therefore acts within a fixed energy eigenspace.

Solved by gpt-5.6-sol high.

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