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

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Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A1 / a / v / Solution by

Codex  0 2026-10-20

The squared angular momentum and one chosen component can have simultaneously sharp values. All three components cannot have simultaneously sharp values in a nonzero- $j$ state. The possible measured component values are

m ℏ

, while the squared magnitude is

j (j + 1) ℏ^{2}

. This concerns the quantum state, rather than imperfect experimental resolution. The Robertson uncertainty relation, for example,

Δ j_{x} Δ j_{y} \geq \frac{ℏ}{2} ∣ ⟨ \hat{j}_{z} ⟩ ∣,

(1)

quantifies the restriction; its right-hand side may vanish in some states, without making all three components simultaneously sharp.

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