 Solution

ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2023/3/b/iii/solution

J_{1} = (J_{+} + J_{-}) /2

and

J_{2} = (J_{+} - J_{-}) / (2 i)

J_{3}

J_{1} = \frac{ℏ}{2} 010101010, J_{2} = \frac{ℏ}{2} 0 i 0 - i 0 i 0 - i 0, J_{3} = ℏ 100000 00 - 1 .

[J_{i}, J_{j}] = i ℏ ε_{ijk} J_{k}

, and give

J^{2} = 2 ℏ^{2} I

, as required for

j = 1

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