How did we come to know that electrons actually 'move' in an atom? Rutherford's experiments confirmed the existence of light-weight electron clouds in a mostly empty atom, and that they occupy some space around the nucleus. What made us conclude that they can move? Can't it be vice versa: nucleus moves around the electron cloud?
 A: As John and I sais, the electrons in the atom don't move in the sense that they follow some trajectory and have at any time a position and a velocity. We don't really understand how they behave in the atom. Though, they move.
Please see a simple calculus: let's take the hydrogen atom with the electron on the lowest level, n = 1. The wave-function is very simple, spherically symmetrical, proportional to $e^{-r/a_0}$, where $a_0$ is the Bohr radius.
Let's calculate the average linear momenta $P_x$, $P_y$, and $P_z$. Since the wave-function is real,
$<P_x> = -i\hbar \int \psi(r) \frac {∂\psi(r)}{∂x} d\vec r = 0$,
$<P_y> = -i\hbar \int \psi(r) \frac {∂\psi(r)}{∂y} d\vec r = 0$,
$<P_z> = -i\hbar \int \psi(r) \frac {∂\psi(r)}{∂z} d\vec r = 0$,
where the integral is taken over all the atom. So, on average, no net movement in some direction. However,
$<P_x^2> = -\hbar^2 \int \psi(r) \frac {∂^2\psi(r)}{∂x^2} d\vec r = \frac {\hbar^2}{a_0^2} C^2 \int \frac {x^2}{r^2}e^{-2r/a_0} d\vec r$,
where $C$ is the normalization constant.
A similar calculus can be done for $<P_y^2>$ and $<P_z^2>$, and adding the results,
$<P^2> = \frac {\hbar^2}{a_0^2}$.
Thus, the QM says that the electrons in the atom, move. How they do this movement? We don't exactly understand, given that they don't have trajectories.
About the vice-versa, i.e. the nucleus moving around the electrons - well, movement is relative. But the mathematics if we take the nucleus as moving around the electron would be much more complicated. Usually we take the calculus to the center-of-mass frame. Given the disproportion in mass between the nucleus and the electron, the center-of-mass practically coincides with the nucleus.
