8.3 Scattering Theory
249
where
R α,β (n, n
) =
¯
h 2
2m
∞
j =1
φ j,n (x α )φ j,n (x β )
(E − λ j )
(8.23)
is the (n, n )th matrix element of the reaction matrix.
We must now distinguish between propagating and evanescent modes in the
leads. For propagating modes in the left lead, we have
l
n χ
l
k n
(x) =
a n
√
k n
e
+ik n x
−
b n
√
k n
e
−ik n x ,
(8.24)
where a n (b n ) is the amplitude of the incoming (outgoing) particle wave in the nth
channel of the left lead. For propagating modes in the right lead, we have
r
n χ
r
k n
(x) =
c n
√
k n
e
−ik n x
−
d n
√
k n
e
+ik n x ,
(8.25)
where c n (d n ) is the amplitude of the incoming (outgoing) particle wave in the nth
channel of the right lead. The normalization factor,
1
√
k n
, is chosen so the particle
current is normalized to 1 and the S-matrix is unitary. The minus sign in front of
the coefficients b n and d n explicitly takes account of the fact that the matter wave
undergoes a phase shift of 180 ◦ when it reflects. In Eqs. (8.24) and (8.25), we have
k n =
2mE
¯
h 2 −
nπ
w
2
.
(8.26)
If there are M propagating modes, then n = 1, 2, . . . , M.
The evanescent modes in the leads can be written
α
n χ
α
k n
(x) = A
(e)
n e
−κ n |x−x α | ,
(8.27)
where
κ n =
nπ
w
2
−
2mE
¯
h 2
(8.28)
and A
(e)
n is the amplitude. For evanescent modes, the index n = M + 1, M +
2, . . . , ∞. The effect of evanescent modes on scattering in a ripple waveguide has
been discussed in Akguc and Reichl (2001). For the system discussed there, they
are most important for small energy intervals around the threshold energies where
new channels open and an evanescent mode changes into a propagating mode. In
our subsequent discussion, we will often neglect the effect of evanescent modes.
But this cannot always be done and one must be careful. We will comment on this
when necessary.
249
where
R α,β (n, n
) =
¯
h 2
2m
∞
j =1
φ j,n (x α )φ j,n (x β )
(E − λ j )
(8.23)
is the (n, n )th matrix element of the reaction matrix.
We must now distinguish between propagating and evanescent modes in the
leads. For propagating modes in the left lead, we have
l
n χ
l
k n
(x) =
a n
√
k n
e
+ik n x
−
b n
√
k n
e
−ik n x ,
(8.24)
where a n (b n ) is the amplitude of the incoming (outgoing) particle wave in the nth
channel of the left lead. For propagating modes in the right lead, we have
r
n χ
r
k n
(x) =
c n
√
k n
e
−ik n x
−
d n
√
k n
e
+ik n x ,
(8.25)
where c n (d n ) is the amplitude of the incoming (outgoing) particle wave in the nth
channel of the right lead. The normalization factor,
1
√
k n
, is chosen so the particle
current is normalized to 1 and the S-matrix is unitary. The minus sign in front of
the coefficients b n and d n explicitly takes account of the fact that the matter wave
undergoes a phase shift of 180 ◦ when it reflects. In Eqs. (8.24) and (8.25), we have
k n =
2mE
¯
h 2 −
nπ
w
2
.
(8.26)
If there are M propagating modes, then n = 1, 2, . . . , M.
The evanescent modes in the leads can be written
α
n χ
α
k n
(x) = A
(e)
n e
−κ n |x−x α | ,
(8.27)
where
κ n =
nπ
w
2
−
2mE
¯
h 2
(8.28)
and A
(e)
n is the amplitude. For evanescent modes, the index n = M + 1, M +
2, . . . , ∞. The effect of evanescent modes on scattering in a ripple waveguide has
been discussed in Akguc and Reichl (2001). For the system discussed there, they
are most important for small energy intervals around the threshold energies where
new channels open and an evanescent mode changes into a propagating mode. In
our subsequent discussion, we will often neglect the effect of evanescent modes.
But this cannot always be done and one must be careful. We will comment on this
when necessary.