Quantum of Conductance

This isn't a full length review, but it's a quick and neat way to show the quantum of conductance. This is the fundamental amount for a single channel. So for example, if we were to consider a one dimensional wire, no wider than one or two channels, then we'd expect the conductance of the wire to be this value or multiples of it. We begin with the definition of the conductance, which is simply current over voltage.

and the definition of current, which is the amount of charge passing through some cross sectional area in time.

The voltage can be written in terms of the energy needed to move a charge.

Plugging these back into the first equation, we have the following.

Now we draw on the Heisenberg Uncertainty principle for energy and time, and

with an extra factor of two to account for the spin degeneracy, we have the result!