 Solution

ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/section-b/9/e/solution

Use the dimensionless coordinate

u = x / L

P = \int_{0.45}^{0.55} 2 sin^{2} (2 π u) d u = [u - \frac{s i n 4 π u}{4 π}]_{0.45}^{0.55} .

Therefore

P = 0.1 - \frac{sin ( π /5 )}{2 π} = 0.00645 ≃ 0.645%.

The central interval surrounds a node, which explains why the probability is much smaller than its ten-percent share of the well's length.

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