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?
 A: 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.
A: Most people would say that $|\Psi(a)|^2$ is the probability density that the particle would be found in some small neighborhood of $a$.
Most people would not say it is the probability that it is in that location. This is because because that same kind of thinking (that it has a probability of having a property and that a measurement merely reveals the value of that property that already existed) will literally get you in trouble if used in other situations. For instance a spin measurement objectively does not reveal a preexisting value of a component of a vector. And in fact a spin measurement objectively changes the thing you "measure."
But there is a tradition that does do so, and it found a way to make that work for position and it is called the de Broglie-Bohm (dBB) interpretation (or the pilot wave theory). The price to be paid is that if you say the probability is accurate for an actual position then other so called measurements can't be merely revealing preexisting values.
This is because the measurements can be correlated with position so knowing the position forces the position to determine everything else (and the context and calibration of the exact set up can be a factor too and again this is because those can correlate to position).
If we used different words this wouldn't sound so strange. We could call them spin polarizations instead of spin measurements for instance.
And in the pilot wave theory trajectories are not meaningless. However they aren't what you observe. You observer through interactions. So all you know is the interactions, you don't know the trajectory.

In that sense, how should we understand the motion of a particle in Quantum Mechanics?

There is a notion of probability current (or probability current density). It is, for instance, how we compute the fraction of a beam that reflects or transmits through a barrier. You can imagine streamlines that have that current as their tangents. And for the pilot wave theory a particle has a location and it follows one of those streamlines. And for everyone else it is just one streamline of many.
Almost no one uses pilot waves since the trajectories are much more detailed than is needed just to compute what fraction of your results come out a certain way. But they can still talk about the probability current and that is usually what they mean if they talk about a particular direction of motion. They just don't literally think there is a particle with a location. They are just talking about current.
So you can have locations and if you do then you need to update them. Most people don't imagine locations but still do talk about the current, they just call it a probability current instead of imagining a probability of being located somewhere and then that updating in time.
And you would have a problem if you thought there were x y and z components of a spin vector and that the interactions we call "measuring the z component of spin" merely passively revealed that value. So if thinking they don't exist for position reminds them to not think they exist for components of a spin vector then great. It is great to learn not to make mistakes. And you can still compute all the frequencies you need without imagining particles having actual locations and moving around.

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

You can not worry about it and stick to probability currents, they will track the flow of probability which is all you need. Or you can study pilot wave theory, which doesn't help you make any new predictions and some of those trajectories are weird. For instance the wave function is defined on configuration space so it is the configuration that changes and so they can be highly nonlocal in many senses.
A: A quantum system is described by a suitable $*$-algebra of observables. The quantum states are functionals of the observables, that when applied on observables yield the average value of it in the state. So, given an observable $A$, and a state $\omega$, $$\omega(A)$$ is the evaluation of $A$, i.e. its average value on the state.
Now the state (or equivalently the observables) evolve in time. This means that we have a map $\omega_{(\cdot)}$ that describes how the state changes with time. In other words,
$$\omega_t(A)$$
would describe the average value of $A$ at time $t$. If we choose the position operator $x$ as an observable, we get a function of time that we may call effective (averaged) trajectory:
$$\bar{x}(t)=\omega_t(x)\; .$$
This averaged trajectory may be interpreted as a notion of motion of a QM particle. Obviously, $\bar{x}(t)=x_0$ it is not a good notion to say that a particle is static (the particle may be moving but be on average on the same place); but a non-constant function $x(t)$ is a good indication of the motion of a QM particle.
In addition, the classical trajectory is actually $\bar{x}(t)$, in a regime where the quantum effects become negligible.
A: To deal with the concept of motion in Quantum Mechanics, we must start with the well-known incompatibility problem inherent in the fusion of quantum mechanics and special relativity. One of the main examples that can illustrate this incompatibility is the space-time diagram, in which well-defined world lines are used to represent the paths of elementary particles, while quantum mechanics forbids the existence of such world lines.

To harmonize quantum mechanics and the theory of relativity, we are forced to assume that a free particle wave function exists at any point in both space and time.
In order for a particle to appear on the diagrams, it needs to be localized through interaction in a certain region of space-time. After such localization, the wave function of the particle again spreads both in space and time. Thus, the motion consists of a successive series of localizations along the trajectory of the particle.
The main conclusion: the movement in the classical sense is applicable only for a wave packet, which is localized in space and time to some extent. For it, one can accurately determine the particle velocity --- the group velocity of the packet, and as a consequence --- construct a world line. The condition of such a movement is
\begin{equation*}
    \frac{\partial |\psi|^2}{\partial t} \neq 0.
\end{equation*}
Let's consider some example.
For the Hydrogen atom ground state, we have the solution:
\begin{equation*}\label{}
     \psi = \sqrt\frac{1}{\pi}\frac{1}{a^{3/2}}e^{r/a}e^{\frac{i}{\hbar} E t},
\end{equation*}
where $a$~--- the Bohr radius, $E = -\frac{me^2}{2\hbar^2}$.
In suche case $\frac{\partial |\psi|^2}{\partial t} = 0$. Again, did we get motion? No, of course, we can only find out where the electron can be with one or another probability. Yes, the electron is not moving, since it is in a stationary state. And of course, the paradox with the radiation of an electron is solved in quantum mechanics right off the bat --- since the electron does not move, it means that it does not radiate.
I'll send you a quote of Max Born from his "Atomic Phycisc"

From another point of view, the statistical interpretation of wave
functions suggests how the radiation emitted by the atom may be
calculated on wave-mechanical principles. In the classical theory this
radiation is determined by the electric dipole moment fi of the atom,
or rather by its time-rate of variation. By the correspondence
principle,  this connexion must continue to subsist in the wave
mechanics. Now  the dipole moment $\mathbf{p}$ is easily calculated by
wave mechanics; if we  adhere to the analogy with classical atomic
mechanics, it is given by
\begin{equation*}
     \mathbf{p} = e \int \mathbf{r} |\psi|^2 dv =e \int \mathbf{r} \psi_n^*\psi_n dv \end{equation*}
where r stands for the radius vector
from the nucleus to the, point oJ  integration, or field point. (As
usual, the asterisk denotes replacement  by the conjugate complex
quantity.) The integral represents of" course  the position of the "
electrical centroid of the electronic cloud". Now  as is easily
proved, this integral vanishes for all states of. an atom  so that the
derivative of the dipole moment vanishes, and according!}  the emitted
radiation also; that is, a stationary state does not radiate.

A: In classical physics we have always dealt with the motion of non-punctual objects as fluids and rigid bodies (let’s call them fields). The trajectory of those objects is clearly not a single line. However, by experience we notice that such fields have special symmetry points were all other points seem to move around. These points are known as the center of mass (or better, center of energy) and therefore, the trajectory of a field can be decomposed into two. One is the clear path of the center of mass and other is motion of every individual point around it. In quantum mechanics I always view particles as fields just as in classical mechanics. You can call these as probability density fields $\Psi(p)=|\psi(p)|^2$, but I prefer to call them THE “particle distribution”. Such a distribution governs the distribution of every other property of the particle.

Single point interation pattern of particles. By Dr. Tonomura and Belsazar
Now, if particles are volumetric, then why particle interaction with a screen builds up as single points like in the image above? Well, by the same reason one can empty an entire bathtub thru a single hole. In fact, we could drain an entire ocean from a small hole, if we could make that hole down to the center of earth and create some room there for it. Fluids have 2 characteristics that make them prone to do that: 1) If you remove some of the fluid in a particular point, disturbing they natural shape, it brings fluid from the vicinity to restore its original shape; 2) The bigger the gradient (removal) the faster it pours into that point.
Characteristic 2) is important because, it means that a fast passing particle will have a bigger change to pour all of its energy into a particular point before getting too far from it, if the value  of its energy distribution is high enough at that point to allow for a big uptake. This means that the higher the value of the “fluid distribution” in a particular point, the higher the probability for the particle to strongly interact in that point, i.e. $\Psi(p)$ gives the probability density that a particle will (strongly) interact into a particular point.
If you view particles this way, many problems of quantum physics become trivial. Let me give you some examples:

*

*The double slit experiment: particle passed thru both holes with those causing a reshape of the particle from $\Psi_1$ to $\Psi_2$. For example, from spherical symmetry into peaks and valleys, causing the interaction pattern in the screen to change from a located small dot into an interference pattern;

*The uncertainty principle: You cannot find a particle in an infinitesimally small volume. The smaller the volume you integrate the “particle distribution”, the bigger the chunk of the particle you miss;

*The non-irradiation nature of standing electrons in an atom: Electrons (particles) can trade shape for an equivalent amount of energy (and momentum) just as you can do by reshaping a spring or a rubber band. They can trade “moving energy” (kinetic energy) into an equivalent “standing energy” (potential energy). For example, the electron cloud in the orbitals, are the electron themselves and they are not moving. If they move from one shape into another, this means that they will have to eliminate some momentum and energy equivalent to the difference of these quantities from the old to the new setup.

Nevertheless, there are 2 points that need to be clarified:
First, why particle seems to always interact in a single point, even when $\Psi$ have equally high values at multiple points? To answers this, we need to understand that in order to have a strong interaction, we need a sink (other particles, fields, etc). Due to the chaotic nature and instantaneous existence of those in a particular point the interaction happens for the first one that gathers all necessary things first. In other words, in a particle sinking contest, the winner takes it all.
Second and more importantly, what is the nature of that “particle distribution” I mention before? For that question I don’t have an answer. One needs to remind himself that $\psi$ is a spinor field. There is no equivalent to that in classical physics, so maybe it is just out of our intuition to imagine a quantity like that, but, for now, the probability wave nature of $\psi$ seems to work, and I don’t want to rock the boat.
In summary:

*

*Particles have defined trajectories that correspond to their “center of distribution’s” path, even though they may not strongly interact in that point, as it could be in a valley. This is a similar case to that of a hollow rigid sphere in classical physics;

*It seems that high energy free particles tend have their strong interaction probability concentrated around their “center of distribution” as I pointed out in this question, causing them to mostly behave as a particle with clear trajectories rather than a wave.

