Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2019 / Paper 2 / Section B / 21 / iv / b / Solution

For a wire of length $L$, one spin state has $Ldk/(2\pi )$ states in one propagation direction. Including two degenerate spin states and using $dϵ/dk=\hslash v$ givesDivide by $L$ for density per unit length. The factor-of-two distinction is essential: the current from one reservoir uses only its incoming propagation branch. If spin degeneracy is lifted, count each resolved spin channel separately.