Flux carried by a directed electronic mode ($d\dot{N}=dE/h$)

For one directed spin-resolved channel, the density of longitudinal states per unit length and wave number is $1/(2\pi )$. Since $v={\hslash }^{-1}dE/dk$, velocity times density per energy is $1/h$. A reservoir occupation imbalance therefore contributes particle flux $\int \delta f(E)dE/h$; in zero-temperature linear response it is $\delta \mu /h$. Multiplying by charge and transmission gives the Landauer current per mode.