QMGreensite_merged

144 CHAPTER9. THEHARMONICOSCILLATOR

oscillator,however,thiscommutatorisverysimple:

[A,B] =

√

1

2 m





√

1

2

kx^2 , ̃p





=

√

k

4 m

[x,p]

= i ̄h

√

k

4 m

(9.19)

Inclassicalphysics,theresonantangularfrequencyofaharmonicoscillatorofmass
mandspringconstantkis

ω=

√

k

m

(9.20)

so

[A,B]=i

1

2

̄hω (9.21)

Therefore

H ̃=C†C+^1

2

̄hω (9.22)

and [
C,C†

]

= ̄hω (9.23)

WenowdefinetheLoweringOperator

a =

√

1

̄hω

C

=

√

1

̄hω





√

k

2

x+i

p ̃

√

2 m





=

1

√

2 ̄h

(

√

mωx+i

p ̃

√

mω

)

(9.24)

anditsHermitianconjugateknownastheRaisingOperator

a† =

√

1

̄hω

C†

=

1

√

2 ̄h

(

√

mωx−i

p ̃

√

mω

)

(9.25)

Thecommutatorofthesetwooperatorsfollowsfrom[C,C†]= ̄hω,

[

a,a†

]

= 1 (9.26)