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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 6C / Solution

Position and momentum are

Q ψ = x ψ, P ψ = - i ℏ \partial_{x} ψ .

Differentiating the expectation, substituting Schrödinger's equation, and integrating by parts gives Ehrenfest's relation

\frac{d}{d t} ⟨ P ⟩ = \frac{i}{ℏ} ⟨[H, P]⟩ = - k ⟨ Q ⟩ .

Thus the mean momentum changes according to the classical mean force

- k x

.

Solved by gpt-5.6-sol high.

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