Movement of free electrons

The movement of free electrons through a conductor is quite a useful visualization for passage of current through a conductor but nevertheless, probing this observation on the basis of quantum mechanics and using Heisenberg's uncertainity would be a non-simplifying endeavour unless one clears the classical picture on the first go.

First of all, free electrons actually don't go anywhere.If you have been habituated to describing an image of infinite microscopic spheres which enter a conductor at one end and leave through the other, disabuse yourself of the notion to proceed further.

Secondly, one must concretely define what a general definition of current is. An ideal definition follows from a version of Noether's theorem which states that: To every differentiable symmetry generated by local actions there corresponds a conserved current. The symmetry is an invariant 'function' of the action which is an integral of the Lagrangian.

The current can be corresponded to a flow of anything, like something as abstract as probability current for quantum mechanics. In our case conserved current implies charge conservation which, non-obviously follows from the fact that the magnitude of the wavefunction in Schrodinger's equation is independent of the phase which is the invariant here.

So much for description. Now you might want to know the correspondence of the electric current with the above principles. The conventional electric current is merely a response induced in a circuit to oppose the 'gradient'(vector slope of some potential source, also called field ,might be gravitational or in this case, electric). This is quite similar to a ball rolling down a slope except for some non-trivialities described below.

Even from classical calculations it is instructive to observe that electrons could not carry energy at all.The free electrons, seen like an electron gas if you will (fermion gas, NON-IDEAL), have or rather are supposed to have random motion or more correctly random distribution through the conductor. Application of the electric field induces a small drift velocity to the electrons opposite to the field through the wire. This drift velocity is Orders of Magnitude Smaller than speed of effect of electric current like turning on a bulb ! So, how to explain this ?

The answer lies in the fact that the energy in the circuit is carried by electric and magnetic fields both of which are perpendicular to the wire. For ideal non resistive wires,to a degree of simplification the electric field is not along the wire but perpendicular to it and parallel to the field established by the voltage source while the B-field is perpendicular and have circular field lines surrounding it. The cross product of these two,gives the Poynting vector which consequently gives the direction and magnitude of energy flow and it is parallel to the non-resistive wire till it reaches the load where it points to the interior of the resistor indicating energy consumption like light from an electric bulb. The wires can be said to be merely acting as waveguides in a sense.

Hence current can be said to be an attempt by the circuit to redistribute the charge from the voltage or rather energy source. The function of an energy source is to produce non-equilibrial variations in the corresponding fields at the cost of its internal energy (when it cannot,it dies out, this is why batteries die). Chnges across the field are transmitted near instaneously and this is the 'speed of electricity'.

So when you connect a source, say it supplies some energy. From quantum mechanics you might know that uncertainity compels atoms and subatomic electrons to be seen as charge 'clouds' (Heisenberg in play here due to position and momentum uncertainity). What the source does is polarise the clouds or more simply alter the shape of the charge clouds of atoms throughout the wire. From Gauss' law, no charge can exist in the interior of a conductor, so the net effects of polarisation are seen on the conductor surface which in turn produces an electric field outside the conductor surface despite no net charge in the conductors (this is the reason while transmission lines despite being electrically 'neutral' has capacitance with respect to the earth and would fry you if you touched them). The clouds of course try to return to the initial shapes with same centre for positive and negative charge clouds but this is not possible due to the source which maintains this condition along the conductor surface.

Lastly for open circuits there is no Poynting vector despite being connected to a source because of the lack of magnetic field which arise from continuous current flow. Current continuity is an important reason why no energy flow takes place.

Long answer but hope this clears the question. Any comments or criticism would be welcome and appreciated.

Ηow can I understand the movement of free electrons in a conductor taking into account the quantam mechanical approach of electrons i.e. uncertainty of position and momentum etc.

Here is a "visualization" of the current in a conductor, for example. Quantum mechanics with its probabilistic underlying axioms, cannot be "visualized". There are no "free" electrons as everything is bound in a quantum mechanica wavefunction of great complexity. What one can do is to use simplified quantum mechanical models and "visualize" the corresponding solutions to the problem at hand.

For example , the band theory of solids accepts that the orbitals of the electrons partake of the geometry of the whole solid and are different energy levels:

There the electrons you call "free" are in the conduction band. while the rest are bound to the lattice locations.

Does using quantum mechanics changes the way we understand the motion electrons in a conductor ?

Yes, it becomes a probability of motions, according to the solutions of the quantum mechanical problem.