A quantum transport framework that relates electrical conductance of phase-coherent mesoscopic conductors to the transmission probabilities of electronic scattering channels between reservoirs, yielding conductance as a sum over transmission eigenchannels (at zero temperature G = (2e^2/h) Σ_n T_n for spin-degenerate, two‑terminal coherent transport).

Principle

Conductance is governed by quantum transmission: macroscopic resistance emerges from partial transmission and reflection of electronic wave modes at scatterers and contacts rather than from a local, bulk resistivity alone.

Demonstration

Illustrative scenario: A narrow quantum point contact between two electron reservoirs supports a small number of transverse modes with transmission probabilities T_n. Recognition: transport is phase coherent and reservoirs thermalize carriers. Action: compute net current from reservoir chemical potential difference using transmission probabilities. Consequence: conductance steps of ΔG = 2e^2/h appear as modes open, and contact/scattering-induced transmission reduction directly lowers G.

Misapplication

Applying the two‑terminal Landauer formula unchanged to systems with strong inelastic scattering inside the conductor, significant many‑body interactions, finite dephasing length shorter than device length, or ill-defined reservoirs; the semantic error is treating single‑particle, coherent transmission as universally applicable without accounting for incoherent processes or interactions.

Consequence

Correct application reinterprets resistance as contact- and scattering-controlled, predicts quantized conductance and sensitivity to transmission channels, and informs device design (mode matching, contact engineering). Misapplication leads to wrong conductance estimates and misattribution of sources of dissipation.

Reversal

When incoherent, diffusive transport or strong many‑body correlations dominate, or when the notion of thermal reservoirs coupled by elastic scatterers is invalid, conductance must be described by semiclassical Boltzmann, Kubo linear-response, or interacting quantum transport approaches rather than the simple Landauer sum.

Boundary

Clearly within: low-temperature, phase-coherent, mesoscopic two‑ or multi‑terminal conductors with well-defined leads and negligible inelastic scattering inside the sample. Boundary case: systems with moderate inelastic scattering that can be modelled by adding fictitious voltage probes (Büttiker probes). Clearly outside: macroscopic diffusive conductors at room temperature where local resistivity and scattering dominate.

Semantic Tension

Single‑particle coherence versus many‑body/incoherent processes — Landauer frames transport as single‑particle scattering, which competes conceptually with bulk resistivity descriptions that average over incoherent scattering.

Synthesis

Landauer–Büttiker reframes electrical resistance as a wave‑scattering and contact problem: counting and weighting transmission channels yields conductance in coherent regimes, and deviations from the formalism diagnose the onset of inelasticity, interactions, or reservoir nonidealities that require alternative transport descriptions.