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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 15B / b / iii / Solution

The phase has unit modulus and

X = x + g t^{2} /2

, hence

⟨ X ⟩_{Θ} = ⟨ x ⟩_{ψ} + \frac{1}{2} g t^{2} .

The free-particle equations imply

⟨ X ⟩ = a + v t

. Therefore

⟨ x ⟩_{ψ} = a + v t - \frac{1}{2} g t^{2} .

The wavepacket centre obeys the classical constant-acceleration trajectory, an instance of the correspondence principle.

Solved by gpt-5.6-sol high.

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