Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 35C / c

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- Solution c

Solution

The required state is the maximal-spin coherent state pointing at azimuth

φ

in the equatorial plane. In the same ordered

S_{z}

basis it is

∣ n, 3/2 ⟩ = \frac{1}{2 2} 1 3 e^{i φ} 3 e^{2 i φ} e^{3 i φ} .

Its norm is

(1 + 3 + 3 + 1) /8 = 1

. Direct multiplication using the matrices in part (b), or rotation of

∣ \hat{z}, 3/2 ⟩

, gives

(S_{x} cos φ + S_{y} sin φ) ∣ n, 3/2 ⟩ = \frac{3}{2} ℏ∣ n, 3/2 ⟩ .

It therefore yields the value

3ℏ/2

with probability one when spin is measured along

n

.

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