256
8 Manifestations of Chaos in Quantum Scattering Processes
S(E) = e
−2ika (1 + ikaR(E))
(1 − ikaR(E))
= e
−2ika e
iθ (E) ,
(8.61)
where
θ
(E) = 2arctan
k
k tan(k
a)
(8.62)
with k =
2mE/ ¯
h 2 and k =
2m(E − V 0 )/ ¯
h 2 . For this example, the coupling matrix,
¯
w, is a 1×N -dimensional column matrix given by
¯
w =
¯
h 2 k
2m
⎛
⎜
⎜
⎜
⎝
φ 1 (a)
φ 2 (a)
. . .
φ N (a)
⎞
⎟
⎟
⎟
⎠
=
¯
h 2 k
2m
2
a
⎛
⎜
⎜
⎜
⎝
1
−1
. . .
−1(1)
⎞
⎟
⎟
⎟
⎠
(8.63)
(for N even (odd)). The matrix ¯
w· ¯
w T is an N ×N matrix with one nonzero eigenvalue,
N ¯
h 2 k
ma . The remaining N − 1 eigenvalues are all equal to zero (Reichl and Akguc 2001).
8.4 Wigner–Smith and Partial Delay Times
One of the most important quantities that can be derived from the S-matrix is the
Wigner–Smith delay time for the system. As Wigner first showed, the Wigner–
Smith delay time is a measure of the length of time that a particle is delayed in
the reaction region (relative to the time it would take to traverse the reaction region
if no reaction processes were present). If we are given a scattering matrix, which
is a unitary matrix, its eigenvalues will be complex numbers that lie on the unit
circle. To each eigenvalue we can associate an eigenphase. The slopes of these
eigenphases, when plotted as a function of energy, are called the partial delay times.
Their average value, at any given energy, is called the Wigner–Smith delay time. In
the neighborhood of energies for which the S-matrix has complex energy poles, the
delay times can become very large. In the subsections below, this behavior will be
made explicit using the reaction matrix theory described in Sect. 8.3.
8.4.1 Delay Time of a Wave Packet
The meaning of the delay time is most easily seen by looking at the scattering of a
wave packet in one space dimension. We assume that a wave packet, incident from
the right, is reflected by a hard wall at x = 0 and a potential barrier, V (x), such that
V (x) > 0 for 0≤x≤x o and V (x) = 0 for x > x o . We will assume that the wave
8 Manifestations of Chaos in Quantum Scattering Processes
S(E) = e
−2ika (1 + ikaR(E))
(1 − ikaR(E))
= e
−2ika e
iθ (E) ,
(8.61)
where
θ
(E) = 2arctan
k
k tan(k
a)
(8.62)
with k =
2mE/ ¯
h 2 and k =
2m(E − V 0 )/ ¯
h 2 . For this example, the coupling matrix,
¯
w, is a 1×N -dimensional column matrix given by
¯
w =
¯
h 2 k
2m
⎛
⎜
⎜
⎜
⎝
φ 1 (a)
φ 2 (a)
. . .
φ N (a)
⎞
⎟
⎟
⎟
⎠
=
¯
h 2 k
2m
2
a
⎛
⎜
⎜
⎜
⎝
1
−1
. . .
−1(1)
⎞
⎟
⎟
⎟
⎠
(8.63)
(for N even (odd)). The matrix ¯
w· ¯
w T is an N ×N matrix with one nonzero eigenvalue,
N ¯
h 2 k
ma . The remaining N − 1 eigenvalues are all equal to zero (Reichl and Akguc 2001).
8.4 Wigner–Smith and Partial Delay Times
One of the most important quantities that can be derived from the S-matrix is the
Wigner–Smith delay time for the system. As Wigner first showed, the Wigner–
Smith delay time is a measure of the length of time that a particle is delayed in
the reaction region (relative to the time it would take to traverse the reaction region
if no reaction processes were present). If we are given a scattering matrix, which
is a unitary matrix, its eigenvalues will be complex numbers that lie on the unit
circle. To each eigenvalue we can associate an eigenphase. The slopes of these
eigenphases, when plotted as a function of energy, are called the partial delay times.
Their average value, at any given energy, is called the Wigner–Smith delay time. In
the neighborhood of energies for which the S-matrix has complex energy poles, the
delay times can become very large. In the subsections below, this behavior will be
made explicit using the reaction matrix theory described in Sect. 8.3.
8.4.1 Delay Time of a Wave Packet
The meaning of the delay time is most easily seen by looking at the scattering of a
wave packet in one space dimension. We assume that a wave packet, incident from
the right, is reflected by a hard wall at x = 0 and a potential barrier, V (x), such that
V (x) > 0 for 0≤x≤x o and V (x) = 0 for x > x o . We will assume that the wave