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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 15D / a / Solution

The Schrodinger equation and its complex conjugate are

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

Because the potential

U

is real, its two contributions cancel when differentiating the probability density

ρ = ψ^{*} ψ

. Therefore

\frac{\partial ρ}{\partial t} = ψ^{*} \frac{\partial ψ}{\partial t} + ψ \frac{\partial ψ ^{*}}{\partial t} = \frac{i ℏ}{2 m} (ψ^{*} \nabla^{2} ψ - ψ \nabla^{2} ψ^{*}) = - \nabla \cdot [- \frac{i ℏ}{2 m} (ψ^{*} \nabla ψ - ψ \nabla ψ^{*})] .

The expression in square brackets is the probability current

J

, so

\frac{\partial ρ}{\partial t} + \nabla \cdot J = 0 .

This probability continuity equation says that probability can leave a region only through the current across its boundary.

Solved by gpt-5.6-sol high.

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