Solution
= Solution
Under the stated potential transformation, set
$$
\psi'(x,t)=e^{-ief(x,t)/\hbar}\psi(x,t).
$$
Then
$$
\left(\nabla+\frac{ie}{\hbar}(A+\nabla f)\right)\psi'
=e^{-ief/\hbar}\left(\nabla+\frac{ie}{\hbar}A\right)\psi
$$
and
$$
\left(\partial_t-\frac{ie}{\hbar}(\phi-\partial_tf)\right)\psi'
=e^{-ief/\hbar}\left(\partial_t-\frac{ie}{\hbar}\phi\right)\psi.
$$
Applying the spatial identity twice proves the <Gauge covariance of the Schrödinger equation>.