Digital Signal Processing Reference
In-Depth Information
see that in the Wigner domain both the mathematics and insight become
clearer. We point out that the results we present for the Schr odinger equation
are classic in the work of Wigner, Moyal, and many others, but the results
we present for the diffusion equation we believe to be new. We mention that
these equations may be related to the respective Fokker-Planck equations,
but that will not be pursued here.
We point out that the equation for diffusion with drift is
∂
u
∂
c
∂
u
D
∂
2
u
t
+
x
=
x
2
,
(4.47)
∂
∂
and the respective Wigner equation of motion is
2
W
u
∂
∂
W
u
∂
c
∂
W
u
∂
D
2
∂
2
Dp
2
W
u
+
=
−
.
(4.48)
t
x
x
2
However, no generality is lost by taking the drift term equal to zero because if
u
(
x, t
)
solves the no-drift equation, Equation 4.38, then
u
(
x
−
ct, t
)
will solve
the equation with drift. Similarly, if
W
u
(
x, p, t
)
satisfies Equation 4.46, then
W
u
(
x
−
ct, p, t
)
satisfies Equation 4.48.
4.5.1
Green's Function
4.5.1.1 Schr odinger Equation
Suppose we want to solve the initial value problem for the Schr odinger equa-
tion. That is, given
ψ(
)
ψ(
)
>
x,
0
we want
x, t
,
where
t
0. The solution is
x, x
,
0
x
,
0
dx
,
ψ(
x, t
)
=
G
ψ
(
)ψ(
)
(4.49)
x, x
,t
where
G
ψ
(
)
is the Green's function:
exp
−
(
−
x
)
2
1
x
x, x
,t
G
ψ
(
)
=
√
4
.
(4.50)
4
iat
π
iat
In momentum space the initial value problem becomes particularly easy. From
Equation 4.41 we have
e
−
iap
2
t
φ(
p, t
)
=
φ(
p,
0
).
(4.51)
Now consider the same problem for the Wigner distribution, that is, given
W
(
x, p,
0
)
we want
W
(
x, p, t
).
From Equation 4.45 it follows that
W
ψ
(
x, p, t
)
=
W
ψ
(
x
−
2
apt, p,
0
)
,
(4.52)
a result first obtained by Wigner and Moyal. Thus, a remarkable simplification
is achieved in phase space. But furthermore in phase space we understand
what is going on. It shows that as time progresses the phase space point moves
with a constant velocity in the
x
direction but does not move at all in the
p
direction. The velocity in the
x
direction is 2
ap
.
Search WWH ::
Custom Search