We consider an idealized QPC model: the potential is zero everywhere except along a hard, thin wall where it is taken as infinite. (This corresponds to the reflection phase shift ). The `scattering system' is an aperture (hole) in this wall, of width centered at the origin (see Fig. 7.1b). The desired solutions to the Schrodinger equation in this system are solutions of the Helmholtz equation corresponding to Dirichlet boundary conditions on the wall. This is a paradigm system for the study of classical particle conductance, and because of the sharp edges it forms a very non-adiabatic point contact. However, in mesoscopic 2DEG systems where depletion regions are defined electrostatically by gates the potential cannot possess structures sharper than about due to screening[20,55]. Therefore as a model of real 2DEG systems, it is approximate.
In the limit of a small aperture ()
the transmission cross-section scales
for constant , and is dominated
by p-wave (dipole) scattering.
This is derived in Appendix K, including the correct prefactor.
The result is
The limit of a large aperture ()
is the semiclassical limit, and the
conductance is that of a classical
(first studied in 3D by Sharvin .)
The cross section is a purely classical quantity
(independent of ), being
simply the projection
of the aperture length
onto the incident beam
The exact result at arbitrary is also known. We refer the reader to Chapter 11.2 of  and  for the detailed form of the wavefunctions. They can be expressed as Mathieu functions  in elliptical coordinates defined by and . The Mathieu function expansion of the incoming wave is also needed. Calculation of requires summing the transmitted flux of the lowest few transverse Mathieu functions .
We apply this scheme to generate Fig. 7.2a, which shows the cross section for arbitrary width of the aperture and for various incident angles . Fig. 7.2b shows the angular average (our plot differs from that of  only in that we show cross section as a fraction of the normal-incidence classical cross section ). This plot also shows asymptotic convergence to the small- and large- results presented above, and gives an idea of when breakdown occurs.
The oscillations in have the same period as the quantization steps in an adiabatic QPC of the same minimum width, but are much weaker, even though we are at zero temperature: this is because our QPC is very nonadiabatic [189,206].