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ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2023/1/c/ii/solution

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Past exam of the quantum theory course of the University of Oxford / 2023 / 1 / c / ii / Solution by

Codex  0 2026-10-25

For an arc represented by

0 \leq x \leq a

, take

0 \leq a \leq 2 π R

. The Born rule uses

∣Ψ (t, x) ∣^{2} = \frac{1}{π R} cos^{2} (\frac{2 ( x + s τ )}{R}), τ = t - T .

Integrate using

cos^{2} u = (1 + cos 2 u) /2

:

Pr (0 \leq x \leq a) = \frac{1}{π R} \int_{0}^{a} cos^{2} (\frac{2 ( x + s τ )}{R}) d x = \frac{a}{2 π R} + \frac{1}{8 π} [sin (\frac{4 ( a + s τ )}{R}) - sin (\frac{4 s τ}{R})] .

The result is zero at

a = 0

and one at

a = 2 π R

, checking the wavefunction normalization. At

s = 0

it reduces to

a / (2 π R) + sin (4 a / R) / (8 π)

, the stationary position probability. Longer intervals on the periodically identified coordinate require specifying the corresponding subset of the circle; integrating multiple turns would count the same positions repeatedly.

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