In a metallic conductor, during the electric current, do the electrons flow from one end to the other or they just vibrate

Question

Do the electrons cover the entire path of a circuit during electric current or they vibrate or oscillate at their positions?

Answer 1

They move.

In the case of an DC current with a battery as the source, there is an electrochemical reaction inside it. Positive ions migrate from the anode to the cathode. It is necessary a flow of electrons through the external circuit also from the anode to the cathode for the reaction continues.

In the case of AC current they move back and forth. But if by vibrate you mean a very small displacements, in the range of one atomic distance, it is not true. That is the behaviour of electrons in insulators, not in conductors.

Answer 2

One can answer this question on two levels. The so-called classic picture would be that electrons move with rather high velocity in all directions, even in the absence of a current. They bump with each other, with the ions and with the sides of the conductor. A current is simply the slight bias of velocities in one direction.

In the quantum picture electrons are not small balls. They are extended objects (i.e. wavefunction) with some distribution of momenta. A current imply the distribution of momenta is biased.

So we cannot say that a specific electron had travelled across the entire pass.