Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 34E / b

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- Solution b

Solution

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.

Multiply the Schrodinger equation by

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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