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# Appendix L: Explicit relation of cross section to Landauer formula

In this appendix we derive the relation (7.23) formally in a self-contained fashion. No reference to the semiclassical derivation of Section 7.1 is made. Therefore this provides an independent derivation of our main result (7.1).

We first present the special case corresponding to a QPC system embedded in a hard wall (at ). The reflection off the unbroken wall is diagonal in the half-plane partial-wave basis of Section 7.2. We can expand the unscattered wave' in Bessel functions [continuing the expansion of Eq.(7.20)], to give

 (L.1)

Recognising this as being composed entirely of a sum of the half-plane partial-wave bases (7.13) on the left side, we can read off the coefficients . (Note that the outgoing coefficients are also equal to this). With the QPC system now placed in the wall, the outgoing amplitudes on the right side are . This gives a flux
 (L.2)

Using the definition of cross section (7.4) we can write
 (L.3)

from where the orthogonality of the radial functions allows us to replace the final integral by , whereupon the sum becomes a trace, giving
 (L.4)

completing the proof.

Now considering a general wall profile with a general , the situation becomes more subtle. An attempt to expand in Bessel functions as in (L.1) gives additional terms which are not part of the half-plane basis (7.13), namely an (s-wave) term and terms with angular dependence for all . This would suggest that the basis (7.13) is incomplete. However, all these new terms can be expressed as sums of the functions already present, if we are careful to consider the large radius limit before the maximum angular momentum limit . In this limit , a given basis state will have negligible -wavevector near the wall, compared to the wall profile width, so the softness' of the wall profile will not be apparent. Rather, the wall will appear as a hard reflector at . We write the entire left side wavefunction

 (L.5)

in a general form where the incoming and outgoing angular distributions are apparent. Therefore, even for general , the apparent hardness of the wall forces and to go smoothly to zero as . This ensures that the original basis set is sufficient to represent all asymptotic wavefunctions, since , form a complete set in the interval . The set , form another complete set in the same interval; both sets are not required and we are able to choose just one [7]. Because of the hard wall boundary condition in -space, the former set is more appropriate.

We are still left with the issue of finding the incoming coefficients given an unscattered wave for general . The problem is subtle, but can be understood when we consider the order of limits above: restricting allows the incoming representation of a single plane wave to become a well-defined, narrowly-peaked delta-like function about the incident angle. This single plane wave will also cause a similar delta peak in the opposite direction, which it turns out is irrelevant because it contributes instead only to outgoing . Thus we have the important result that we can ignore the reflected wave (making its phase shift irrelevant) in calculating , because this wave can only contribute to in the interval . We can use stationary-phase integration (method of steepest descents) [7] applied to a single plane wave to show this, and find the delta strength, as follows. Taking care with the definitions of angle (see Fig. 7.1a) we have

 (L.6)

where . Expanding the cos as a quadratic about the stationary point , and making the usual stationary-phase replacement
 (L.7)

gives . The other stationary point results in the same delta strength but an additional factor , which turns the incoming to an outgoing , thus contributing only to . Generally, we can write the expression for incoming partial-wave expansion
 (L.8)

which is easily derived using orthogonality of the angular functions and the asymptotic form of . Substituting the narrowly-peaked resulting from the single incident plane wave gives
 (L.9)

These are the same incoming partial wave coefficients as derived above for the case , which is as expected since the reflected wave phase has not entered into our considerations. Thus our proof (L.4) holds for general .

Next: Bibliography Up: Dissipation in Deforming Chaotic Previous: Appendix K: Transmission cross
Alex Barnett 2001-10-03