From uncertainty to certainty in quantum conductance of nanowires

We show using the uncertainty principle that the conductance per channel in the quantum ballistic transport regime is G≤2e2/h. Then, relying on the phase space arguments when density of states is a valid description, we convert this inequality into equality without invoking contact resistance explicitly. Our ability to derive the quantum conductance formula using the Heisenberg uncertainty principle explains why many diverse calculations such as first principles electronic structure, molecular dynamics, and even Luttinger liquid based calculations readily arrive at the same result. We find that the integer quantum Hall effect and quantized Hall conductance follow from similar considerations. We then apply the uncertainty principle argument to derive a value for quantum electronic thermal conductance in nanowires. It is noted that this value differs from the value in the bulk system by a numerical constant. Consequently, the Wiedemann–Franz law in the quantum domain also differs from the bulk value by a numerical factor. It will be interesting to see if experiments can detect this departure in ballistic transport through nanowires.