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)