 Solution

ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2021/1/c/iii/solution

For a position measurement, the Born rule integrates the probability density over the chosen interval. The three terms from part (b) have integrals

\int_{0}^{π /4} 1 d x = \frac{π}{4}, \int_{0}^{π /4} cos^{2} (2 x) d x = [\frac{x}{2} + \frac{s i n 4 x}{8}]_{0}^{π /4} = \frac{π}{8},

and

\int_{0}^{π /4} 2 cos (2 x) cos (Ω t) d x = [sin (2 x) cos (Ω t)]_{0}^{π /4} = cos (Ω t) .

Thus

Pr (0 \leq x \leq \frac{π}{4}) = \frac{1}{8} + \frac{1}{3 π} cos (\frac{2ℏ t}{m}) .

The oscillating term comes from quantum interference between the zero and positive energy components. The probability lies between

1/8 - 1/ (3 π)

and

1/8 + 1/ (3 π)

, both within

[0, 1]

; integrating the same probability density over the whole circle gives one.

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