8.4 Wigner–Smith and Partial Delay Times
257
packet has a small momentum spread, k, with most probable value at k = k 0 . We
can write the wave function in the asymptotic region as
ψ(x, t) =
B
√
ππk
∞
−∞
dk exp
−(k − k 0 ) 2
((k) 2
(e
−ikx
− S(E)e
ikx )e
−i ¯
hk 2 t/2m ,
(8.64)
where B is a normalization constant, E = ¯
h 2 k 2 /2m, S(E) = e iθ(E) is the scattering
function for this system, and θ(E) is the phase shift of the scattered particle. The
minus sign in front of S(E) is due to the hard wall and accounts for a phase shift of
π that always takes place upon reflection from a hard wall.
Let us expand the phase shift, θ(E), about the most probable value of the
eigenvector, k = k 0 ,
θ(E) = θ(E 0 ) +
dθ
dk
k=k 0
(k − k 0 ) +
1
2
d 2 θ
dk 2
k=k 0
(k − k 0 )
2
+ . . . .
(8.65)
If we keep only the lowest-order contributions to the phase shift, we can perform
the integration in Eq. (8.64), and we find
ψ(x, t) = e
−i ¯
hk 2
0 t/2m
B
√
ππk
π
in (k 0 , t)
e
−ik 0 x exp
−(x + ¯
hk 0 t/m) 2
4 in (k 0 , t)
−
π
out (k 0 , t)
e
+ik 0 x exp
−(x − ¯
hk 0 t
m + ¯
h 2 k 0
m
dθ
dE
k=k 0
) 2
4 out (k 0 , t)
,
(8.66)
where in (k 0 , t) =
1
((k) 2 +
i ¯
ht
2m is a measure of the spread of the incident wave
packet and out (k 0 , t) =
1
((k) 2 +
i ¯
ht
2m −
1
2
d 2 θ
dk 2
k=k 0
is a measure of the spread of
the scattered wave packet. We have also used the fact that
dθ
dk
k=k 0
= ¯
h 2 k
m
dθ
dE
k=k 0
.
Equation (8.66) describes the behavior of the wave packet in the asymptotic region
(away from the scattering potential). The incoming part of the wave packet has
its maximum value (and therefore maximum probability of finding the particle) at
points where x = − ¯
hk 0
m t for t < 0. The outgoing part of the wave packet has its
maximum value at points where x = ¯
hk 0 t
m − ¯
h 2 k 0
m
dθ
dE
k=k 0
. Thus, there is a delay
time,
τ ≡ ¯
h
dθ
dE
k=k 0
,
(8.67)
of the scattered particle due to its interaction with the scattering potential, V (x)
(Bohm 1950).