130_notes.dvi

10 Harmonic Oscillator Solution using Operators

Operator methods are very useful both for solving the Harmonic Oscillator problem and for any type
of computation for the HO potential. The operators we develop will also be useful in quantizing the
electromagnetic field.

The Hamiltonian for the1D Harmonic Oscillator

H=

p^2

2 m

+

1

2

mω^2 x^2

looks like it could be written as the square of a operator. It can be rewritten in terms of the operator
A(See section 10.1)

A≡

(√

mω

2 ̄h

x+i

p

√

2 m ̄hω

)

and its Hermitian conjugateA†.

H= ̄hω

(

A†A+

1

2

)

We will use the commutators (See section 10.2) betweenA,A†andHto solve the HO problem.

[A,A†] = 1

The commutators with the Hamiltonian are easily computed.

[H,A] = − ̄hωA

[H,A†] = ̄hωA†

From these commutators we can show thatA†is a raising operator (See section 10.3) for Harmonic
Oscillator states

A†un=

√

n+ 1un+1

and thatAis alowering operator.

Aun=

√

nun− 1