 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-1/a1/a/iii/solution

[A, BC] = [A, B] C + B [A, C]

[\hat{j}_{z}, \hat{j}_{x}^{2}] [\hat{j}_{z}, \hat{j}_{y}^{2}] [\hat{j}_{z}, \hat{j}_{z}^{2}] = i ℏ (\hat{j}_{y} \hat{j}_{x} + \hat{j}_{x} \hat{j}_{y}), = - i ℏ (\hat{j}_{x} \hat{j}_{y} + \hat{j}_{y} \hat{j}_{x}), = 0.

The first two expressions cancel on addition, so

[\hat{j}_{z}, \hat{j}^{2}] = 0.

The same argument works for each component of the angular momentum operator.

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