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The equations involving drift speed, mobility and conductivity, $v_d=\mu E$ and $\sigma=\mu e n$, treat electrons in a Newtonian way. This model works well in practice (at least in macro scale), but I wonder what is actually happening to an electrons during "drift"? In particular, is it just a massive number of tunnelling and hopping events?

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Transport of this kind is a terribly complicated quantum many body problem.

Electrons in metals are problematic, because they form a degenerate Fermi gas (or rather a Fermi liquid), so classical descriptions do not work well at all (in contrast to ions in solutions, where classical descriptions are not too bad approximations). The ground state of such a system is a Fermi sea, where all states up to a maximal momentum are occupied.

The electron eigenstates in (Fermi liquid) metals are rather states of definite crystal momentum than states of definite position. Therefore, the image of an electron hopping from lattice site to lattice site as a transport mechanism is flawed. It is better to image electron wave packets to move with the group velocity given by the momentum derivative of the dispersion relation of the band.

The problem here is, that without scattering processes electrons in a crystal will oscillate if a DC voltage is applied (because the dispersion relation is periodic in $\vec k$-space, the keyword for an intuitive picture of this is Bragg reflection).

One common way to handle such transport problems mathematically is the Boltzmann equation. The problem is, that the Boltzmann equation depends on the notion of a time and space dependent momentum distribution function for the electrons, this quantity however is not well defined in general (but works well if the distribution changes sufficiently slow in space).

Now each momentum state corresponds to a "wave packet" travelling with its group velocity. If the occupation is symmetrical (as it is in equilibrium), there will be no current (as there is a wave packet with exactly the opposite intrinsic movement, so right and left moving currents cancel out). If we now apply DC voltage scattering processes equilibrate the occupation of the possible $\vec k$ states to be asymmetrical. Therefore, there is an imbalance between currents to (say) the right and the left, and so a net current develops.

So in conclusion, you cannot decompose quantum transport in a "series of events". A less crude picture is to consider the transport of wave packets through the crystal. To do so properly you have to consider the Boltzmann equation that describes the system by the distribution reached by "intrinsic" transport and scattering mechanisms. In the end all transport is just the motion of (quasi-)particles and as these are quantum objects their motion is described by the motion wave packets.

Disclaimer: While the picture offered here is superior to the classical picture given by the Drude theory, it is nevertheless only a semi-classical approximation, that cannot account for systems where quantum-correlations are important. The discussion also ignores inter-band effects (tunnelling between bands, this becomes important in the case of the Zener current, where electrons from a full valence band tunnel to the empty conduction band).

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