In this appendix we derive the transmission cross-section
of the idealised slit from Section 7.3, in the
tunneling limit .
We first consider a simple scaling argument which gives the dependence on
and
, and then find the correct prefactor.
The slit system possesses a scaling property (shared by all hard-walled
quantum systems):
it is invariant under a rescaling of space
provided there is a corresponding rescaling of wavevector
.
Our slit potential is defined by a single parameter
, the slit
half-width.
The particle wavelength
is the only other length scale in the problem.
This means that for a given incident wave,
the form of the full scattering wavefunction,
and hence the ratio of the quantum to classical transmission cross-sections,
can only depend on
.
(In other words, if the wavelength changes in proportion to the slit size,
we are viewing the original problem with a new unit of length.)
The classical cross-section is proportional to
.
Therefore the quantum cross-section must have the exact form
We seek in the
limit.
For a hard-walled system solutions of Schrodinger's equation become
those of the Helmholtz equation with Dirichlet boundary conditions.
Close to the slit (a distance
)
they become
well approximated
by the
limit, namely solutions of Laplace's equation.
Thus we can use `electrostatics' at these small distances,
and then match to the small-
tails of the
Helmholtz solutions to find the transmitted flux.
The solutions obtained in this way have been known for over a century,
and were first found by Lord Rayleigh ([168]
in which the scattering solutions are derived for
Dirichlet
and Neumann apertures
and their inverses, in both 2D and 3D.
See p. 270-1 for our particular case).
The transmission is dominated by scattering of incoming dipole (
)
radiation to outgoing dipole radiation; we will justify
this shortly.
The dipole parts of the 2D Laplace solution can be written generally as
![]() |
(K.2) |
The dimensional scaling of (K.3) implies ,
so at constant
the transmitted flux, and hence
, grows like
.
Comparison with (K.1) immediately gives
the scaling
.
This
dependence of cross section for a given aperture size is
the 2D equivalent of Rayleigh's famous
law for dipole light scattering in 3D explaining why the sky
appears blue [167] (see also [105] p.418).
How does the contribution of higher multipole radiation scale?
The exact Laplace solution with general asymptotic form
will involve generalizing
Eq.(K.3) to a linear matrix relation connecting all the
and
with
to all those
with
.
However because of the associated
and
factors,
by dimensional arguments
all the elements of this matrix (other than the
already considered
above)
will scale like
powers of
higher than 2.
It is important to realise that the Laplace coefficients
(
) directly connect to the respective
coefficients of Bessel functions of
type for
and
type for
on the left (right) sides, which
in turn determine the incoming and outgoing waves of angular
momentum
on those same sides.
Therefore the resulting contribution to
due to
excitation by the
channel on the left and re-radiation into the
channel on the right
will scale as
.
This contribution
will therefore be smaller than the dipole-dipole
(
,
) component by a factor
.
In fact, because of symmetry about the
-axis, the
system only couples even
to even
and odd
to odd
;
this guarantees that the most significant correction
is in fact a factor
down.
Thus in our
limit we are justified
in using only p-wave scattering: dipole `absorption' from
,
and dipole re-radiation into
.
(There will also be equal dipole re-radiation into
,
which interferes with
but does not affect the conductance).
We now can find the prefactor.
The connection between the Laplace dipole solution ()
and the outgoing
dipole coefficient
(see Section 7.2)
is found by matching to the small-
form
[7] of the Neumann part
of the Hankel function.
This gives
.
Combining this with (K.3) and the correct Laplace slit result
[168,148]
gives
![]() |
![]() |
![]() |
|
![]() |
![]() |
(K.5) |
Note that
is equal to
, because
the reflected dipole strength is equal to the transmitted.
(The dipole is also `two-headed', radiating in phase on both sides).
It is interesting to note that in any hard-walled
scattering system,
the conductance can only be a function of the product .
This follows from substitution of Eq.(K.1)
into Eq.(7.12).