$i\hslash$). Time $t$ is an external parameter upon which the state vectors depend, which accounts for why Eq. (\[ak\]) specifies $t$ as a label independent of the Hilbert space. A unitary transformation preserves this equivalence. This can be stated as$$i\hslash\partial_{t}-\widetilde{H}=U\left(  i\hslash\partial_{t}-H\right)
U^{-1}. \label{al}$$ Since Eq. (\[al\]) can be written as$$\widetilde{H}=UHU^{-1}+U\left(  i\hslash\partial_{t}U^{-1}\right)  ,
\label{am}$$ this shows explicitly that the Hamiltonian operator does not transform unitarily if there is any time dependence in $U$.

An important gauge transformation is that introduced by Göppert-Mayer [@gm], widely employed in the AMO community. This transformation is given by$$U_{GM}=\exp\left(  \frac{ie}{\hslash c}\mathbf{r\cdot A}\left(  t\right)
\right)  , \label{an}$$ which depends explicitly on time when $\mathbf{A}\left(  t\right)  $ describes a laser field within the dipole approximation, meaning that Eq. (\[am\]) is consequential.

The fact that, in general,$$\widetilde{H}\neq UHU^{-1}, \label{ao}$$ is the explanation for the curious result to be found in many papers (for example, Refs. [@yang; @kobesmirl; @beckerss; @lss; @jbauer]) that the $\mathbf{r\cdot E}$ potential is a preferred potential. If any other potential is employed in solving the Schrödinger equation, then the claim is made that a transformation factor must be employed even on a non-interacting state. There is a logical contradiction inherent in the requirement that a non-interacting 