Operators in Heisenberg and Schrodinger pictures I don't understand the difference between the Schrödinger picture and the Heisenberg picture in quantum mechanics. Here's some of my doubts:

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*If in the Heisenberg picture state vectors are constant in time and in the Schrödinger picture operators are constant in time, in which picture am I if I study a system where the potential is time dependent, such as the interaction between an Hydrogen atom and radiation? In this particular example, the Schrödinger equation is
$$i\hbar \dfrac{\partial}{\partial t}\Psi(\vec{r},t)=\left(  -\dfrac{\hbar^2}{2m}\nabla^2 -\dfrac{e^2}{4\pi\varepsilon_0r} -\dfrac{i\hbar e}{m}\vec{A}(\vec{r},t)\cdot \vec{\nabla} +\dfrac{e^2}{2m}A^2(\vec{r},t) \right)\Psi(\vec{r},t) $$
where $\vec{A}(\vec{r},t)$ is the vector potential. Here both the wave function and the operators which contain the vector potential are time dependent.


*According to this wikipedia page https://en.wikipedia.org/wiki/Heisenberg_picture an operator in the Heisenberg picture satisfies the equation
$$\frac{d}{dt}A_\text{H}(t)=\frac{i}{\hbar}[H_\text{H},A_\text{H}(t)]+\left( \frac{\partial A_\text{S}}{\partial t} \right)_H,$$
where the $H$ and the $S$ indicate the operator in the Schrödinger or in the Heisenberg picture. In this equation appears the time derivative of an operator in the Schrödinger picture, which I don't understand since operators in the Schrödinger picture are time independent.
 A: The operator equation here has close parallels in Hamilton equations of motion in classical mechanics (up to replacement of the commutator by a Poisson bracket with appropriate coefficient):
$$
\frac{d}{dt}f(p,q,t)=\frac{\partial}{\partial t}f + \{f,H\},
$$
which also distinguish the explicit time dependence of the operators from their inherent time dependence due to the Hamiltonian evolution.
Thus, regaring the point 1. - we have to distinguish the explicit time dependence that appears, e.g., in operator $\mathbf{A}(\mathbf{r},t)$ even in the Schrödinger picture, and the implicit time dependence that will arize only in the Heisenberg picture.
Regarding the second point - I believe that the subscript $S$ in the term with the partial derivative is an error. In fact the equation is a closed equation for the Heisenberg operators.
I would suggest working first through simple examples, such as free particle:
$H=\frac{p^2}{2m}$ and harmonic oscillator $H=\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}$,w here the equations for the operators $x,p$ can be explicitly written and solved in the Heisenberg picture.
