7.B.7. Incompressible Flow
Using
X N  pN , qN 
eq(7.15) can be rewritten as
X N  X N  0
(7.21)
which means the probability density flow is incompressible.
The balance equation (7.8) then simplifies to
 N
 X N  X N  N  0
t
(7.22)
The total or convective time derivative
d

  X N  X N
dt t
(7.23)
measures the time rate of change in a frame moving with the velocity X N of the
probability fluid. Eq(7.22) is therefore simply
dN
0
(7.24)
dt
so that the probability density is a constant in a frame that moves with the same
velocity as the fluid.
Reverting to the p, q notation, we have


X N  X N  p N  N  q N  N
p
q

H

H

 N  N  N
N
q p
p q
 HN
so that (7.22) can be rewritten as
 N
 HNN  0
t
(7.26)
(7.25)
Setting the (hermitian) Liouville operator as
LN = i H N
we have
 N
(7.27)
 LN  N
t
which is called the Liouville equation.
i
For a stationary state,
 N
 0 , so that
t
LN  N  0
(7.29)
The formal solution to (7.26) is simply
 N  t   e  iL t  N  0 
N
(7.28)