Example: spatially constant electric field, zero magnetic field

We choose a gauge (set $c=1$)

$$\phi=0, \mathbf{A}= - \int_{-\infty}^t dt' \mathbf{E}(t').$$ (2.9)

We transform to a new gauge

$$\mathbf{A}' = \mathbf{A}+ \nabla f=0, \quad \phi' = \phi -\partial_t f= -\partial_t f$$

$$\leadsto \nabla f = - \mathbf{A},\quad f({\bf r},t) = {\bf r} \int_{-\infty}^t dt' \mathbf{E}(t') \leadsto \phi'({\bf r},t) = -{\bf r}\mathbf{E}(t)$$

$$\leadsto i\partial_t \psi'= H'(t)\psi' \quad H'(t) = \frac{p^2}{2m} -q{\bf r}\mathbf{E}(t).$$ (2.10)