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

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

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

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

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

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

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

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 .