That is in a zero electric field, if an electron moving, does it follow rectilinear propagation? Does it follow Newton's first law of motion?
I not asking about newtonian mechanics. I am asking considering HUP and QM, if I shoot an electron will it go straight? Is the velocity constant? This happens to be newtons 1 law. It was asked on behalf of all tiny quantum particles, especially electron.
Electrons macroscopically act as zero point charged massive particles . Macoscopically they follow all the classical laws, including Newtons.This is an experimental fact, otherwise how could one have accelerators and colliders.
The beams collide within 64 microns at LHC. The protons follow the classical solutions both for Newton and electromagnetism until collision with another proton, where quantum mechanics takes over.
For an electron in the quantum regime (here defined as a particle whose position and momentum uncertainties $\Delta x$ and $\Delta p$ have a produce $\Delta x \, \Delta p$ that is of the order of $\hbar$), Newton's first law still holds in a quantum sense (basically, if there are no forces on the particle then its momentum distribution will not change), but the Heisenberg Uncertainty Principle demands that the particle must form a wavepacket with a finite momentum uncertainty $\Delta p>0$, and that uncertainty in the velocity means that the wavepacket will spread in position space.
Generally speaking, though, if the wavepacket isn't "too quantum" (i.e. its propagation does not involve too much interference between different components of the matter wave), then this spreading will be describable using liouvillian mechanics, i.e. the classical mechanics of a particle with an uncertain position and momentum, where an ensemble of particles with uncertain momentum will still spread in position even though the newtonian First Law still holds.
It's also important to note that your understanding of Newton's laws is rather flat, and that there is a lot of nuance there that you're completely trampling over. A proper understanding of Newton's laws will re-shape them quite a bit from the form in which they're presented at high-school level; the re-shaped version is explained in detail in this answer, and in that scheme the first law reads
First Law. Local inertial reference frames exist.
In that form, the first law is absolutely still valid in QM, and indeed it is a core part of the background that allows QM to work. The rest of Newton's laws, however, simply don't work, because they talk about the trajectory of a given particle and in QM particles simply do not have trajectories, and it is counter-factual to try and speak about them. To be crystal clear, that means that when you say things like
Now assume that we introduce an electron in the sphere, as slowly as possible
which assume that the electron has a definite position in a QM context, then you're already colouring so far outside the lines that the argument is already wrong from that point onwards.
Electrons kinematical properties of course obey Newton dynamics, but you have to take into account, that even single electron alone is charged particle.
If it is under influence of any force ( and in fact the only possible forces here are fundamental quantum fields: electroweak or gravitation) it get some acceleration, and then radiates electromagnetic fields. That's why single electron in classical regime is not described by Newton kinematics alone ( as for example a thrown stone), but by theory of electrodynamics in classical limit, where number of particles is constant.
You may imagine that radiated electromagnetic field take some energy from electron. That's why electrons are hard to accelerate, because this radiation factor grows very rapidly as function of velocity. Magnetic and electric fields generated by electron move, transforms itself exactly in a way predicted by Maxwell-Clark equations. Even when there is no acceleration, constant velocity only, electromagnetic field still is present, which is deeply connected to special relativity theory ( and the first paper Einstein wrote about SRT was titled "On the move of electrodynamics of moving bodies"). ( Edited unfortunate words about radiation).
