= Solution
In <quasi-one-dimensional quantum transport>, transverse confinement makes discrete subbands while motion along a <quantum wire> remains continuous. Only occupied propagating subbands conduct. For elastic scattering between equilibrated reservoirs, the <Landauer formula> is
$$
\boxed{G=\frac{e^2}{h}\sum_{n,\sigma}\int dE\,[-f'(E)]T_{n\sigma}(E).}
$$
The sum explicitly counts spin. A perfectly transmitting spin-degenerate subband gives $2e^2/h$ at low temperature; a <quantum point contact> shows steps as a gate opens additional subbands. The one-dimensional <density of states> cancels the carrier velocity in the current, giving a conductance quantum rather than a material-dependent velocity factor. Finite two-terminal <contact resistance> remains even without internal scattering.
Transport is <ballistic transport> for length $L\ll\ell$, where $\ell$ is the <mean free path>. In a multichannel wire with $\ell\ll L$ below its localization length, scattering reduces transmission and gives approximately Ohmic behaviour. If the <phase coherence length> exceeds the sample length, amplitudes interfere; <weak localization> and sample-dependent <universal conductance fluctuations> can appear. At greater lengths, generic coherent disordered wires undergo <Anderson localization>, with localization length of order the channel number times $\ell$, up to symmetry-dependent factors. Thermal averaging and dephasing smear the quantum features.
Strong interactions require more than independent-electron transmission. A clean single-channel interacting wire is described at low energy by a <Luttinger liquid>, with collective modes and possible <spin-charge separation>; impurities can generate temperature-dependent power laws. Ideal contacts to noninteracting leads can still fix the two-terminal conductance of a clean wire, so it must not automatically be multiplied by a bulk Luttinger parameter.
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