How can one describe electron motion around hydrogen atom? I remember from introductory Quantum Mechanics, that hydrogen atom is one of those systems that we can solve without too much ( embarrassing ) approximations.
After a number of postulates, QM succeeds at giving right numbers about energy levels, which is very good news.
We got rid of the orbit that electron was supposed to follow in a classical way ( Rutherford-Bohr ), and we got orbitals, that are the probability distribution of finding electron in space.
So this tiny charged particle doesn't emit radiation, notwithstanding its "accelerated motion" ( Larmor ), which is precisely what happens in real world.
I know that certain "classic questions" are pointless in the realm of QM but giving no answers it makes people asking the same questions over and over. 


*

*If the electron doesn't follow a classic orbit, what kind of alternative "motion" we can imagine?

*Is it logical that while the electron is around nucleus it has to move in some way or another?

*Is it correct to describe electron motion as being in different places around nucleus at different instants, in a random way?

 A: The problem is that you're thinking of the electron as a particle. Questions like "what orbit does it follow" only make sense if the electron is a particle that we can follow.
But the electron isn't a particle, and it isn't a wave either. Our current best description is that it's an excitation in a quantum field (philosophers may argue about what this really means; the rest of us have to get on with life). An electron can interact with its environment in ways that make it look like a particle (e.g., a spot on a photographic plate) or in ways that make it look like a wave (e.g., the double slits experiment) but it's the interaction that is particle-like or wave-like, not the electron.
If we stick to the Schrödinger equation, which gives a good description of the hydrogen atom, then this gives us a wavefunction that describes the electron. The ground state has momentum zero, so the electron doesn't move at all in any classical sense. Excited states have a non-zero angular momentum, but you shouldn't think of this as a point like object spinning around the atom. The angular momentum is a property of the wavefunction as a whole and isn't concentrated at any particular spot.
A: You might be helped by reading carefully the wikipedia article on the hydrogen atom particularly the figures.
The electron described in the orbital has not only a specific energy but also momentum and angular momentum, though it is only the operators of energy angular momentum and spin that give the eigenvalues for n l and m.
So what is random is not the electron per se but the probability of finding it when you try to measure it in some way . It is moving with 1/137 of the velocity of light according to the linked article.

If the electron doesn't follow a classic orbit, what kind of alternative "motion" we can imagine?

as given in the pictures of the orbitals. such a fast moving particle will look like a cloud anyway, even if possible classically.

Is it logical that while the electron is around nucleus it has to move in some way or another?

Yes, we just cannot pin it, think of the uncertainty principle organized by a solution to  Schrodinger's equation.

Is it correct to describe electron motion as being in different places around nucleus at different instants, in a random way?

No, not random. It is organized by the probabilities of the orbital it happens to be in.
A: 
Is it logical that while the electron is around nucleus it has to move in some way or another?

That probably depends of what exactly you call motion, but I would highly recommend an excellent book And Yet It Moves by Mark P. Silverman, and the chapter #3 in particular. If you replace an electron (which is a stable particle, that is a particle without age and individual history) in a simple atom with a negative muon (which decays quickly, its lifetime being some 2 microseconds in its rest frame) you would expect that measured lifetime (in the atom or lab rest frame) will be longer if the muon moves at relativistic velocities due to time dilation, exactly as experiments confirm.
A: Think of an electron as of non-point particle. In a hydrogen atom it is "smeared" around proton. Its total momentum is zero – it is neither moving (in total) nor accelerating – hence in a classical limit it does not emit radiation.
If an electron in an atom is a "cloud" rather than a point, it is in different points at the same time. That means that there is a non-zero distribution of "electron density" smeared around proton.
An electron is not "moving" as a whole, but we can say that "parts of the cloud" are moving, since they carry non-zero momentum resulting in total angular momentum. This is a consequence of the fact that integration of the electron's momentum density over limited volume in space is non-zero.
A: Some of the statements above need a little bit of correction:
(1) the electron in the groundstate of a hydrogen atom is moving in a radial range around the Bohr radius (most probable distance according to the radial probability density curve resulting from the Schrödinger wave equation).
(2) the average momentum (velocity)  is indeed zero, but its square
<p²> is not! Thus the kinetic energy is nonzero and is purely radial
because the angular momentum and its square (angular kinetic energy) equals zero.
(3) The consequence of (1) and (2) can only be that the electron performs a very fast radial oscillation at some angular position (which may have any value from 0 to 2 pi), without radiation (as energy does not change).
Strange enough, this logical conclusion which is not in contrast to quantum mechanics, is never mentioned; at least I could not yet find it in scientific papers or physics textbooks.
In my opinion, such a trajectory looks plausible and is not in conflict with the results of the wave equation.
