Superposition of Quantum Harmonic Oscillator Eigenstates: Expectation Values and Uncertainties

quantum numbers

n

1

0

n

2

0

complex coefficient

c

0.25

θ

2π

0.25

This Demonstration studies a superposition of two quantum harmonic oscillator eigenstates in the position and momentum representations. The superposition consists of two eigenstates

ψ(x)=

1-c

2



ϕ

n

1

(x)+c

θ

e

ϕ

n

2

(x)

, where

ϕ

n

(x)=

1/4

mω

πℏ

1

n

2

n!

H

n

(

mω/ℏ

x)

-mω

2

x

(2ℏ)

e

and

H

n

(y)

is the

th

n

Hermite polynomial; the representations are connected via



ψ

(p)=

1

2πℏ

∞

∫

-∞

ψ(x)

-px/ℏ

e

dx

. 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 probability density





ψ

(p)

2



, momentum expectation value





p

=

∞

∫

-∞





ψ

(p)

2



pdp

, and momentum uncertainty

Δp=



2



p

-

2





p



. The lower two panels show the real and imaginary parts of the wavefunction.