 Solution

ID: past-exam-of-the-quantum-physics-course-of-the-university-of-oxford/2016/8/a/solution

Hermiticity of the Cartesian angular momentum components gives

L_{+}^{†} = L_{-}, L_{-}^{†} = L_{+} .

[L_{i}, L_{j}] = i ℏ ϵ_{ijk} L_{k}

imply

[L^{2}, L_{\pm}] = 0, [L_{z}, L_{\pm}] = \pm ℏ L_{\pm} .

For example,

[L_{z}, L_{x} + i L_{y}] = i ℏ L_{y} + ℏ L_{x} = ℏ L_{+}

. The two terms in

[L_{x}^{2} + L_{y}^{2} + L_{z}^{2}, L_{i}]

[L_{+}, L_{-}] = 2ℏ L_{z}

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