QMGreensite_merged

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)