How to understand the motion of a particle in Quantum Mechanics?

In Classical Mechanics when we talk about the motion of a particle it is the same as talking about the idea of trajectory. The fact is that in Classical Mechanics, a particle has a definite position given by a point $a\in \mathbb{R}^3$ while in Quantum Mechanics the best we can get is a probability amplitude $\psi : \mathbb{R}^3\to \mathbb{C}$ of the particle presence around a particular point $a\in \mathbb{R}^3$, i.e., such that $|\psi(a)|^2$ is the probability density that the particle is located on some small neighborhood of $a$.

Because of that in Quantum Mechancis the idea of trajectory is meaningless. But nonetheless a particle move around, otherwise it would stay always where it is. In that sense, how should we understand the motion of a particle in Quantum Mechanics?

For instance, I've seem some books talking about "a particle propagating from left to right along the $Ox$ axis" or "a particle comes from the left" when talking about potential barriers. This is of course connected to the idea of motion of the particle, but since we don't have trajectories, I don't know how to understand those statements.

In that setting, how should one understand intuitively and mathematically the idea of motion of a particle in Quantum Mechanics?

Answer

Having worked with elementary particles all my working life I can assure you that particles have a trajectory.

Here is proof

Another proof is the existence of accelerators which create the beams that we can scatter against targets, as in picture, or against each other and study the results statistically. That is how the standard model of particle physics was built up, studying trajectories.

Trajectories are saved by the HUP, the Heisenberg Uncertainty Principle, which given enough momentum can always be fulfilled as the bubble chamber picture shows. Within its bound lie the bound states of atoms , and there we speak of orbitals, not orbits.

Everything is quantum mechanical, and the HUP is the measure of whether classical concepts and mechanics are applicable or not. So in your example the electron can be generated in a small accelerator and approach with a known trajectory a potential well of an ion up to the point of closenes where the indeterminacy of the HUP destroys the concept of a trajectory and the probabilistic form will apply.