WOLFRAM|DEMONSTRATIONS PROJECT

Ladder Operators for the Harmonic Oscillator

​
ω
1
n
0
1
2
3
4
5
6
7
8
9
10
operator
a
†
a
The Hamiltonian for the linear harmonic oscillator can be written
H=-
1
2
2
d
d
2
x
+
1
2
2
ω
2
x
, in units with
ℏ=m=1
. The eigenstates are given by
E
n
=n+
1
2
ω
​
,
ψ
n
(x)=
-1/2
(
n
2
n!)
1/4
(ω/π)
-ω
2
x
2
e
H
n
(
ω
x)
,
n=0,1,2,…
, where
H
n
is a Hermite polynomial. An alternative reformulation of the problem can be based on the representation
H=
†
a
a+
1
2
ω
in terms of ladder operators
a=
ω
2
x+
i
2ω
p=
ω
2
x+
1
2ω
d
dx
​
and
†
a
=
ω
2
x-
i
2ω
p=
ω
2
x-
1
2ω
d
dx
​
. The step-down or annihilation operator
a
acts on the eigenfunctions according to
a
ψ
n
(x)=
n
ψ
n-1
(x)
​
​
, with
a
ψ
0
(x)=0
. The step-up or creation operator
†
a
satisfies
†
a
ψ
n
(x)=
n+1
ψ
n+1
(x)
.
​
In this Demonstration, the eigenfunction
ψ
n
(x)
is plotted in black. Also shown is either
a
ψ
n
(x)
in red or
†
a
ψ
n
(x)
in blue.