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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ii/paper-4/9e/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 9E / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Write

H = F (P, Q)

with

P = \sum p_{k}^{2}

and

Q = \sum q_{k}^{2}

. The angular momentum component

L_{i} = ϵ_{ijk} q_{j} p_{k}

has

\partial_{p_{ℓ}} L_{i} = ϵ_{ij ℓ} q_{j}

and

\partial_{q_{ℓ}} L_{i} = ϵ_{i ℓ k} p_{k}

. Therefore

{H, L_{i}} = 2 F_{Q} \sum_{j, ℓ} ϵ_{ij ℓ} q_{ℓ} q_{j} - 2 F_{P} \sum_{ℓ, k} ϵ_{i ℓ k} p_{ℓ} p_{k} = 0,

since a symmetric product contracts to zero with the antisymmetric Levi-Civita symbol. All three components of angular momentum are conserved by this rotationally invariant Hamiltonian.

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