9.1. RAISINGANDLOWERINGOPERATORS 145
Intermsoftheraisingandloweringoperators,wehave
H ̃ = ̄hω(a†a+^1
2
)
x =
√
̄h
2 mω
(a+a†)
p =
1
i
√
mωh ̄
2
(a−a†) (9.27)
Thereasonforcallingathe”lowering”operatoristhatithastheamazingproperty
oftakinganeigenstateofH ̃ intoanothereigenstate,withalowerenergy. Suppose
φnissomeparticulareigenstate,with
H ̃φn=Enφn (9.28)
Define
φ′=aφn (9.29)
then
H ̃φ′ = h ̄ω(a†a+^1
2
)aφn
= h ̄ω(a†aa+
1
2
a)φn (9.30)
Thecommutationrelation(9.26)implies
a†a=aa†− 1 (9.31)
sothat
H ̃φ′ = ̄hω[(aa†−1)a+^1
2
a]φn
= a[ ̄hω(a†a+
1
2
)−h ̄ω]φn
= a[H ̃−h ̄ω]φn
= (En− ̄hω)aφn
= (En− ̄hω)φ′ (9.32)
whichprovesthat aφnis anenergyeigenstatewitheigenvalueE=En− ̄hω. The
operatora†istermedthe”raisingoperator”forsimilarreasons.Let
φ”=a†φn (9.33)
Then
H ̃φ” = ̄hω(a†a+^1
2
)a†φn
= ̄hω(a†aa†+
1
2
a†)φn (9.34)