Superposition of Quantum Harmonic Oscillator Eigenstates: Expectation Values and Uncertainties
Superposition of Quantum Harmonic Oscillator Eigenstates: Expectation Values and Uncertainties
This Demonstration studies a superposition of two quantum harmonic oscillator eigenstates in the position and momentum representations. The superposition consists of two eigenstates , where (x)=n!( and (y) is the Hermite polynomial; the representations are connected via (p)=ψ(x)dx. The top-left panel shows the position space probability density , position expectation value , and position uncertainty . The top-right panel shows the momentum space probability density , momentum expectation value , and momentum uncertainty . The lower two panels show the real and imaginary parts of the wavefunction.
ψ(x)=(x)+c(x)
1-c
2
ϕ
n
1
θ
e
ϕ
n
2
ϕ
n
1/4
mω
πℏ
1
n
2
H
n
mω/ℏ
x)-mω(2ℏ)
2
x
e
H
n
th
n
ψ
1
2πℏ
∞
∫
-∞
-px/ℏ
e
ψ(x)
2
=ψ(x)xdx
x
∞
∫
-∞
2
Δx=
-
2
x
2
x
(p)
ψ
2
=(p)pdp
p
∞
∫
-∞
ψ
2
Δp=
-
2
p
2
p