Ladder Operators for the Harmonic Oscillator
Ladder Operators for the Harmonic Oscillator
The Hamiltonian for the linear harmonic oscillator can be written , in units with . The eigenstates are given by =n+ω, (x)=(, , where is a Hermite polynomial. An alternative reformulation of the problem can be based on the representation in terms of ladder operators and =x-p=x-. The step-down or annihilation operator acts on the eigenfunctions according to , with . The step-up or creation operator satisfies (x)=(x)
H=-+
1
2
2
d
d
2
x
1
2
2
ω
2
x
ℏ=m=1
E
n
1
2
ψ
n
-1/2
(n!)
n
2
1/4
(ω/π)
-ω2
2
x
e
H
n
ω
x)n=0,1,2,…
H
n
H=a+ω
†
a
1
2
a=x+p=x+
ω
2
i
2ω
ω
2
1
2ω
d
dx
†
a
ω
2
i
2ω
ω
2
1
2ω
d
dx
a
a(x)=(x)
ψ
n
n
ψ
n-1
a(x)=0
ψ
0
†
a
†
a
ψ
n
n+1
ψ
n+1
.
In this Demonstration, the eigenfunction (x) is plotted in black. Also shown is either in red or (x) in blue.
ψ
n
a(x)
ψ
n
†
a
ψ
n