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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ib/paper-4/6b/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 4 / 6B / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

For a normalized wavefunction, the probability density and probability current are

ρ = ∣ ψ ∣^{2}, J = \frac{ℏ}{2 mi} (ψ^{*} \nabla ψ - ψ \nabla ψ^{*}) .

The Schrödinger equation and its conjugate, for a real potential, are

i ℏ ψ_{t} = - \frac{ℏ ^{2}}{2 m} \nabla^{2} ψ + V ψ, - i ℏ ψ_{t}^{*} = - \frac{ℏ ^{2}}{2 m} \nabla^{2} ψ^{*} + V ψ^{*} .

Multiplying respectively by

ψ^{*}

and

ψ

and adding their expressions for the time derivatives cancels the potential terms. Hence

ρ_{t} = \frac{i ℏ}{2 m} (ψ^{*} \nabla^{2} ψ - ψ \nabla^{2} ψ^{*}) = - \nabla \cdot J .

The gradient cross-terms cancel when expanding the divergence. Therefore

ρ_{t} + \nabla \cdot J = 0.

This continuity equation expresses local conservation of probability. Integrating over a region says that its probability decreases by the outward flux of the probability current.

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