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WOLFRAM|DEMONSTRATIONS PROJECT

Probability Densities, Expectation Values, and Uncertainties for Gaussian Wavepackets

​
wavepacket centers
p
0
0
x
0
0
wavepacket width
a
0.3
This Demonstration considers a Gaussian wavepacket
ψ(x)=
1
a
1/2
π
p
0
(x-
x
0
)ℏ
e
-
2
(x-
x
0
)
2
2
a
e
,
ψ
(p)=
1
2πℏ
∞
∫
-∞
ψ(x)
px
e
dx
in the position and momentum representations, respectively. The top-left panel shows the position-space probability density
ψ(x)
2
, position expectation value
x
=
∞
∫
-∞
ψ(x)
2
xdx
, and position uncertainty
Δx=
2
x
-
2
x
. The top-right panel shows the momentum-space probabiity density
ψ
(p)
2
, momentum expectation value
p
=
∞
∫
-∞
ψ
(p)
2
pdp=
∞
∫
-∞
*
ψ
(x)-
ℏ
d
dx
ψ(x)dx
, and momentum uncertainty
Δp=
2
p
-
2
p
. The lower two panels show the complex wavepackets, where the shape is its modulus and the coloring represents the argument (the range
0
to
2π
corresponding to colors from red to magenta).
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