 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/34e/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

D_{A} = \nabla + i e A /ℏ

, the current is

J = - \frac{i ℏ}{2 m} (ψ^{*} D_{A} ψ - ψ (D_{A} ψ)^{*}) .

The two covariant factors acquire opposite phases under a gauge transformation, so

J

is invariant.

ψ^{*}

, subtract its complex conjugate multiplied by

ψ

, and use

ψ^{*} D_{A}^{2} ψ - ψ (D_{A}^{2} ψ)^{*} = \nabla \cdot {ψ^{*} D_{A} ψ - ψ (D_{A} ψ)^{*}} .

The real scalar-potential terms cancel, leaving

\partial_{t} ∣ ψ ∣^{2} + \nabla \cdot J = 0.

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