Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 33B / a / i

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- Solution i

Solution

On the tensor-product Hilbert space, addition of angular momentum gives

J = J_{1} \otimes I_{2} + I_{1} \otimes J_{2} .

Suppressing the identity operators, its components and ladder operators are

J_{x} = J_{1 x} + J_{2 x}, J_{y} = J_{1 y} + J_{2 y}, J_{z} = J_{1 z} + J_{2 z},

J_{\pm} = J_{x} \pm i J_{y} = J_{1 \pm} + J_{2 \pm} .

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