# 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