260
8 Manifestations of Chaos in Quantum Scattering Processes
Fig. 8.5 Plots for
Example 8.3. (a) Phase θ of
S = e iθ . (b) Wigner delay
time. Both plotted as a
function of energy for ¯
h = 1,
V 0 = 10 and a = 1
where k =
(2m/ ¯
h 2 )E and k =
(2m/ ¯
h 2 )(E − V 0 ). The eigenphase, θ(E), is plotted in
Fig. 8.5a for a = 1 and V o = 10.
The delay time is given by
τ (E) = ¯
h
dθ
dE
=
2m
¯
h
1
(k ) 2 + k 2 tan 2 (k a)
×
k
k
−
k
k
tan(k
a) + kasec
2 (k
a)
−
2ma 2
¯
hka
.
(8.78)
The delay time is plotted in Fig. 8.5b for a = 1 and V o = 10. The delay time is negative for
low energies, corresponding to reflection by the barrier before the particle reaches the hard
wall at x = 0. The oscillations that occur at higher energy are due to resonances. As we will
show in Example 8.4, these resonances can be associated with poles of the S-matrix in the
complex energy plane (Reichl and Akguc 2001).
8.4.3 Delay Times and Complex Poles
It is possible to relate the behavior of the Wigner–Smith delay time to complex
poles of the S-matrix. Let us consider the expression for the S-matrix in terms of the
rescaled R-matrix, ¯
K, using Eqs. (8.48) and (8.50). We can write the Wigner–Smith
delay time in the form
τ W S = −
i ¯
h
M
d
dE
ln(Det[ ¯
S]) = −
i ¯
h
M
d
dE
ln
Det[ ¯
1 M − i ¯
K]
Det[ ¯
1 M + i ¯
K]
(8.79)