8.3 Scattering Theory
253
is the rescaled M×M R-matrix (see Eq. 8.37). Note that
¯
B
¯
D
= ¯
U k ·
¯
b
¯
d
and
¯
A
¯
C
= ¯
U
†
k ·
¯
a
¯
c
,
(8.44)
where
¯
a =
⎛
⎜
⎜
⎜
⎝
a 1
a 2
. . .
a M l
⎞
⎟
⎟
⎟
⎠
, ¯
b =
⎛
⎜
⎜
⎜
⎝
b 1
b 2
. . .
b M l
⎞
⎟
⎟
⎟
⎠
, ¯
c =
⎛
⎜
⎜
⎜
⎝
c 1
c 2
. . .
c M r
⎞
⎟
⎟
⎟
⎠
, ¯
d =
⎛
⎜
⎜
⎜
⎝
d 1
d 2
. . .
d M r
⎞
⎟
⎟
⎟
⎠
,
(8.45)
and the matrix ¯
U k is a diagonal unitary matrix with matrix elements
U n,n =
e −ik n x l δ n,n ,
if 1≤n≤M l ;
e +ik n x r δ n,n , if M l + 1≤n≤M.
(8.46)
We now can write
¯
b
¯
d
=
¯
S ll ¯
S lr
¯
S rl ¯
S rr
·
¯
a
¯
c
,
(8.47)
where
¯
S =
¯
S ll ¯
S lr
¯
S rl ¯
S rr
= ¯
U
†
k ·
¯
1 M − i ¯
K
¯
1 M + i ¯
K
· ¯
U
†
k
(8.48)
is the M×M scattering matrix. The scattering matrix, ¯
S, relates the incoming
propagating modes in both leads to the outgoing propagating modes in both leads.
It is usually written in the form
¯
S =
¯
S ll ¯
S lr
¯
S rl ¯
S rr
=
¯
r ll ¯
t lr
¯
t rl ¯
r rr
,
(8.49)
where the M l ×M l (M r ×M r ) submatrix, ¯
r ll = ¯
S ll (¯ r rr = ¯
S rr ), contains probability
amplitudes for reflection (reflection amplitude) into the left (right) lead for waves
that are incident in the left (right) lead. The M l ×M r (M r ×M l ) submatrix, ¯
t lr =
¯
S lr (¯ t rl = ¯
S rl ), contains probability amplitudes for transmission (transmission
amplitudes) into the left (right) lead for waves that are incident in the right (left)
lead.
Let us now consider again the M×M reaction matrix, R n,n , in Eq. (8.23) and
let us keep only the first N (j = 1, . . . , N) of the infinite number of cavity states,
φ j,n (x α ). We can later let N→∞. From Eq. (8.37), we can rewrite the rescaled
reaction matrix K in the form