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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-3/8b/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 8B / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

For a particle of mass

m

, the probability density and one-dimensional probability current are

ρ = ψ^{*} ψ, j = \frac{ℏ}{2 mi} (ψ^{*} ψ_{x} - ψ ψ_{x}^{*}) = \frac{ℏ}{m} Im (ψ^{*} ψ_{x}) .

(1)

The Schrodinger equation and its complex conjugate, for real

V

, give

ρ_{t} = ψ_{t} ψ^{*} + ψ ψ_{t}^{*} = \frac{i ℏ}{2 m} (ψ^{*} ψ_{xx} - ψ ψ_{xx}^{*}) = - j_{x} .

(2)

The potential terms cancel precisely because

V

is real. Thus $ρ_{t} + j_{x} = 0$ , the continuity equation for probability. Integrating over an interval says that its probability changes by the incoming current minus the outgoing current.

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