Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 33C / a

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- Solution a

Solution

The orbital angular momentum operator is

L_{i} = ε_{ik ℓ} X_{k} P_{ℓ} .

Using

[X_{i}, X_{j}] = [P_{i}, P_{j}] = 0

and

[X_{i}, P_{j}] = i ℏ δ_{ij}

,

[L_{i}, X_{j}] = ε_{ik ℓ} X_{k} [P_{ℓ}, X_{j}] = - i ℏ ε_{ikj} X_{k} = i ℏ ε_{ijk} X_{k},

and

[L_{i}, P_{j}] = ε_{ik ℓ} [X_{k}, P_{j}] P_{ℓ} = i ℏ ε_{ik ℓ} δ_{kj} P_{ℓ} = i ℏ ε_{ij ℓ} P_{ℓ} .

Thus the rotation commutators for orbital angular momentum are

[L_{i}, X_{j}] = i ℏ ε_{ijk} X_{k}, [L_{i}, P_{j}] = i ℏ ε_{ijk} P_{k} .

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