Past exam of the physics A course of the University of Cambridge / 2022 / Paper 1 / Section A / 3

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- Solution 3

Solution

The Born rule uses the squared modulus of the wavefunction, not its real part. Write

ψ = A [sin x + i (1 - cos x)]

. Then

∣ ψ ∣^{2} = ∣ A ∣^{2} [sin^{2} x + (1 - cos x)^{2}] = 2∣ A ∣^{2} (1 - cos x) .

Wavefunction normalization requires

1 = 2∣ A ∣^{2} \int_{0}^{2 π} (1 - cos x) d x = 4 π ∣ A ∣^{2}, ∣ A ∣^{2} = \frac{1}{4 π} .

Integrating the probability density over the required interval gives the dimensionless probability

P = \frac{1}{2 π} \int_{0}^{π /2} (1 - cos x) d x = \frac{π /2 - 1}{2 π} = \frac{1}{4} - \frac{1}{2 π} ≃ 0.09085.

Thus approximately $9.08%$ of position measurements fall in this interval.

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