In the context of quantum mechanics, I'm not sure that this is a well-posed question. The idea of "movement" is something whose definition is obvious in an Aristotelian world made up of things like rocks and ice-cream sandwiches, where we have a physical intuition for their position and whether that position changes. But in quantum mechanics, we talk about wavefunctions, which are extended objects.
Let's think about the classic intro-quantum problem of a particle in an infinitely deep square-well potential. If the particle is free (potential $V=0$) on the interval $0<x<L$, your favorite quantum textbook tells you the states are
$$
\newcommand\ket[1]{\left|{#1}\right>}
\newcommand\bra[1]{\left<{#1}\right|}
\newcommand\expect[1]{\left<{#1}\right>}
\newcommand\braket[3]{\left<{#1}\middle|{#2}\middle|{#3}\right>}
\ket n = \psi_n(x) = \sqrt\frac2L\sin\frac{n\pi x}{L}
$$
where $n$ is one of the counting numbers $1,2,3,\cdots$, and each of the states has a distinct total energy proportional to $n^2$, that is, $E_n = n^2 E_1$.
If we measure the position of the particle, our expectation value is
$$
\begin{align}
\expect{x}_n
&= \braket nxn
\\&= \frac2L \int_0^L\mathrm dx\ x \sin^2 \frac{n\pi x}L
\\&= L/2 \quad\text{(for all $n$)}
\end{align}
$$
This result tells us that, if we prepared many such particles and measured their positions, the average of those position measurements would be the middle of the well. But the distributions aren't the same. The variance in the position measurements for the state $\ket n$ is given by
$$
\begin{align}
(\Delta x)_n^2
&= \expect{x^2}_n - \expect{x}_n^2
\\&= L^2 \cdot \left(\frac{1}{12} - \frac{1}{2n^2\pi^2}\right) \quad\text{(after some work)}
\end{align}
$$
meaning that the lower-energy particles are more likely to be found near the center of the well, while the higher-energy particles are spread out more uniformly along the well's length.
So adding energy to a particle in a trap doesn't make the particle move.
Describing the change that does occur requires you to get into some interpretational details.
The most common interpretation of this mathematics is that the state $\ket n = \psi_n(x)$ describes the probability density of locating a zero-sized particle — or at least a particle whose intrinsic size is very much smaller than the other length scales in your problem. But there's an alternate interpretation in which the state $\ket n$ really is all of the information we have about the bound part of the system, and labels like "the particle" are a crutch for those of us who prefer to think about rocks and ice-cream sandwiches. In that sense, increasing the particle's energy doesn't make it move, but it does make the particle wider.
Note in particular that the probability-density interpretation suggests that "the particle" is "really" bouncing around inside of its trap. But the Schrödinger equation actually has a solution which describes a particle bouncing around in its trap. That solution is a superposition of some large number of high-energy/short-wavelength states, which interfere to form a probability density with very narrow $\Delta x$. The eigenstates $\ket n$ that we've given here are "stationary states," whose only change as time evolves is a change in their unobservable complex phase.
