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