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

d

2

x

+

1

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.