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

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- Solution a

Solution

Under the stated potential transformation, set

ψ^{'} (x, t) = e^{- i e f (x, t) /ℏ} ψ (x, t) .

Then

(\nabla + \frac{i e}{ℏ} (A + \nabla f)) ψ^{'} = e^{- i e f /ℏ} (\nabla + \frac{i e}{ℏ} A) ψ

and

(\partial_{t} - \frac{i e}{ℏ} (ϕ - \partial_{t} f)) ψ^{'} = e^{- i e f /ℏ} (\partial_{t} - \frac{i e}{ℏ} ϕ) ψ .

Applying the spatial identity twice proves the Gauge covariance of the Schrödinger equation.

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