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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 15A / v / Solution

With

L = x p_{y} - y p_{x}

, the canonical commutators give

[L, x] = i ℏ y, [L, y] = - i ℏ x, [L, p_{x}] = i ℏ p_{y}, [L, p_{y}] = - i ℏ p_{x} .

It follows that

[L, x^{2} + y^{2}] = 0, [L, p_{x}^{2} + p_{y}^{2}] = 0.

Therefore rotational symmetry of the isotropic oscillator gives

[L, H] = 0 .

Solved by gpt-5.6-sol high.

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