 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2019/paper-1/section-b/b6/a/solution

Write

S_{i} = (ℏ/2) σ_{i}

, with

σ_{i}

σ_{x}^{2} = σ_{y}^{2} = σ_{z}^{2} = I

, and hence

S_{x}^{2} + S_{y}^{2} + S_{z}^{2} = \frac{3}{4} ℏ^{2} I .

For the products in the other order,

σ_{x} σ_{y} = (i 0 0 - i) = i σ_{z}, σ_{y} σ_{x} = (- i 0 0 i) = - i σ_{z} .

[S_{x}, S_{y}] = \frac{ℏ ^{2}}{4} (2 i σ_{z}) = i ℏ S_{z} .

The first identity corresponds to the spin one-half value

s (s + 1) ℏ^{2}

with

s = 1/2

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