WOLFRAM|DEMONSTRATIONS PROJECT

Evolution of a Gaussian Wave Packet

​
momentum
p
0
0
wave packet width
σ
0
1
time
0
A Gaussian wave packet centered around
x=0
at time
t=0
with an average initial momentum
p
0
can be represented by the wavefunction
ψ(x,0)=
-1/2

2π
σ
0

exp-
2
x
4
2
σ
0
+i
p
0
x
. (For convenience, we take
ℏ=m=1
.) The solution of the free-particle Schrödinger equation with this initial condition works out to
ψ(x,t)=
1/2
σ
0
2π
-1/2
2
σ
0
+
it
2
exp-
2
σ
0
2
p
0
exp-
2
x-2i
2
σ
0
p
0

4
2
σ
0
+it2
. The probability density is then given by
ρ(x,t)=ψ(x,t)
2
|
=
-1
(
2π
σ(t))
exp-
2
(x-
p
0
t
)
2
2
σ(t)
, where
σ(t)=
1/2
σ
0
1+
2
t
4
4
σ
0
, shown as a black curve. The wave packet remains Gaussian as it spreads out, with its center moving to
x=
p
0
t
, thereby following the classical trajectory of the particle. The corresponding momentum probability distribution is given by
ρ(p,t)=ϕ(p,t)
2
|
=
2
π
σ
0
-2
2
2
σ
0
(p-
p
0
)
e
, shown in red. The rms uncertainties are given by
Δx=
∫
2
x
ρ(x,t)dx
=σ(t)
,
Δp=
∫
2
(p-
p
0
)
ρ(p,t)dp
=
1
2
σ
0
, which is independent of
t
. This is consistent with the fact that
p
is a constant of the motion for a free particle.
Thus, with ℏ put back in, the uncertainty product is given by
ΔxΔp=
ℏ
2
1/2
1+
2
t
4
4
σ
0
≥
ℏ
2
, in accord with Heisenberg's uncertainty principle. At
t=0
, the Gaussian probability distribution represents a minimum uncertainty wave packet with
ΔxΔp=ℏ/2
, but the product increases when
t>0
.
In this Demonstration, you can drag the time slider to simulate the simultaneous time evolution of the probability and momentum distributions. Note that the
x
distribution broadens with time while the
p
distribution maintains its original width. The numerical values of
x
,
p
, and
t
are illustrative only and have no absolute significance.