Suppose I have a quantum object, inside it the electric field distribution is $\vec{E}(\vec{r})$, with this field we can obtain the scalar potential $\phi(\vec{r})$, a charged particle in this object has the Hamiltonian $H=p^2/2m+q\phi(\vec{r})$. This Hamiltonian will have definite energy levels.
However, if we choose a gauge which represent the electric field by vector field $\vec{A}(\vec{r},t)$, with $\vec{A}(\vec{r},t) = -E(\vec{r})t$, thus $E(\vec{r})=-\partial_t A(\vec{r},t)$. The charge particle's Hamiltonian writes as $H'(t)=(p-qA(\vec{r},t))^2/2m$, which is a time dependent Hamiltonian. In general don't have stationary states solution.
So it seems two gauge choices have contradiction, why?
