 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s3/2/d/solution

Both first moments vanish, so the standard deviations follow directly from the second moments:

Δ x Δ p = \frac{a ^{2}}{7} \frac{5 ℏ ^{2}}{2 a ^{2}} = ℏ \frac{5}{14} ≃ 0.598ℏ > \frac{ℏ}{2} .

The Heisenberg uncertainty relation is satisfied and is not saturated. A Gaussian wave packet can saturate the position-momentum bound; this compactly supported polynomial is not Gaussian.

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