112
J. Aichelin et al.
are facing the problem of how to extend the nonrelativistic QMD approach to the
high energy collisions, considered in this study, within a framework which can be
numerically realized.
In order to extend our approach for relativistic energies, we introduce the modified
single-particle Wigner density ˜
f of the the nucleon i
˜
f (r i , p i , r i0 , p i0 , t) =
(9.13)
=
1
π 3 e
−
2
L (r
T
i (t)−r
T
i0 (t))
2 e
−
2γ 2
cm
L (r
L
i (t)−r
L
i0 (t))
2
×e
−
L
2 (p
T
i (t)−p
T
i0 (t))
2 e
−
L
2γ 2
cm
(p
L
i (t)−p
L
i0 (t))
2 ,
which accounts for the Lorentz contraction of the nucleus in the beam z-direction,
in coordinate and momentum space by inclusion of γ cm = 1/
1 − v 2
cm , where v cm
is a velocity of the bombarding nucleon in the initial N N center-of-mass system.
Accordingly, the interaction density (9.6) modifies as
˜
ρ int (r i0 , t) → C
j
1
π L
3/2 γ cm e
−
1
L (r
T
i0 (t)−r
T
j0 (t))
2
×e
−
γ 2
cm
L (r
L
i0 (t)−r
L
j0 (t))
2 .
(9.14)
With these modifications we obtain
˜
H =
i
p
2
i0 + m 2 − m
+
i
˜
V Skyrme (r i0 , t) +
i
˜
V mom (r i0 , p i0 , t),
(9.15)
with
˜
V Skyrme (r i0 , t) = α
˜
ρ int (r i0 , t)
ρ 0
+ β
˜
ρ int (r i0 , t)
ρ 0
γ
,
(9.16)
and
V mom (p i0 , r i0 ) =
j0 =i0
exp
−c
(p i0 − p j0 ) 2
a(p i0 − p j0 )
2
+ b(p i0 − p j0 )
4
×
⎛
⎜
⎝
C
1
π L
3/2 γ cm e
−
1
L (r
T
i0 (t)−r
T
j0 (t))
2 e
−
γ 2
cm
L (r
L
i0 (t)−r
L
j0 (t))
2
ρ 0
⎞
⎟
⎠ ,
(9.17)
with the time evolution equations (9.2).