One can model the transport of electrons through quantum devices at a number of different levels of sophistication. The simplest level consists of solving the single-particle Schrödinger equation. Because devices are open systems, the solutions of Schrödinger 's equation describing unbounded scattering states are the appropriate basis in which to consider electron transport. Semiconductor heterostructures create some complications in the application of conventional scattering theory, because the electron dispersion relation (band structure) will be non-parabolic and will vary with position. These effects require that the group velocity v be used in most of the fundamental equations of scattering theory where the wavevector k conventionally appears. Extremely robust and efficient numerical techniques have recently been developed which permit evaluation of scattering states, taking into account the these complications.
The near-equilibrium transport properties of a quantum device can be well described with a knowledge of the quantum transmission amplitudes and by invoking the Landauer conductance formula. However, to fully describe the far-from-equilibrium transport on which useful devices depend, one must describe the device in terms of quantum stastical mechanics. Semi-classical formulations, such as the Boltzmann equation and the techniques used to evaluate it, cannot properly deal with quantum interference effects except insofar as they can be described in terms of transition rates. More comprehensive quantum kinetic theories using the Wigner distribution function or non-equilibrium Green's functions have been and are continuing to be developed. To date, however, all of the theories which for which practical computational schemes have been implemented suffer from an inability to deal with one or another of the fundamental conservation laws. Thus there is no one theoretical tool which provides a satisfactory model of all aspects of quantum device behavior.