Explanation of electric current and wave-particle duality of electron Edit note: As I've got a downvote and some negative comments, I will try to make myself very clear.
Electrons are thought to be particles, classically. Electric current is defined to be the movement of electrons.
But, electrons have wave-particle duality.
Q.No:1: How is electric current(defined as movement of electrons,classically) explained with keeping in mind the  wave-particle duality of electrons, as electrons are not really classical particles ?
Q:2: Has QM to do anything with this explanation?
 A: 
How is electric current(defined as movement of electrons,classically)

Incorrect.
Electric current is defined as the motion of electric charge.  In the natural world, most currents are electrolytic, and involve the motion of fairly massive ions both positive and negative.  For example, in acid solutions the electric current is proton-flow.  In saline, the current is Na+ ions moving one way and Cl- ions moving opposite.  In solid metals the electric current is the difference between the motion of conduction-band electrons versus the solid lattice composed of positive metal ions.  (That's why, when we physically move a wire, we're not producing lots of amperes.)  Also, in liquid metals we'll have motion of positive metal ions, not only a pure electron current.
IIRC, I saw the answer to your #2 question some years back.  For electric current in metal wires, the QM effects of a large group of electrons at low average velocity are insignificant.  So, while single/few electrons are quantum objects, an entire "electron sea" or large population of interacting electrons are not.
Similar concept:  charge up a metal ball, then move it along, and you've created electric current on a microamperes scale, but without QM phenomena becoming significant.  Both the crystal lattice and the electron sea of the entire metal ball have well defined position and velocity, and behave like macroscopic materials.  But zoom in on a single metal ion or a single conduction-band electron, and such is not the case.
I recall an interesting/silly experiment from The Physics Teacher magazine.  If we perform a Hall-effect experiment on a large current in a metal strip, performed in order to identify the polarity of the moving charges, and if we then slide this strip along at exactly the opposite of electron-drift velocity, then the results (correctly) show that the moving charge-carriers are positive, not negative.  After all, by moving the wire backwards, we've made the electrons stop drifting wrt the lab frame!  The grid of metal atoms with their positive charges becomes the only "electric current" in the metal.
Knowing the above, it's obvious that a macroscopic copper wire, when moving wrt the stationary electron-sea within the wire, demonstrates an electric current lacking micro-scale QM phenomena.  The same is true if we hold the copper wire still, and only move the electron-sea.
