146 CHAPTER9. THEHARMONICOSCILLATOR
Thecommutationrelation(9.26)alsosaysthat
aa†=a†a+ 1 (9.35)
so
H ̃φ” = ̄hω[a†(a†a+1)+^1
2
a†]φn
= a†[ ̄hω(a†a+
1
2
)+ ̄hω]φn
= a†[H ̃+ ̄hω]φn
= (En+ ̄hω)a†φn
= (En+ ̄hω)φ” (9.36)
andthereforeφ”=a†φnisanenergyeigenstatewitheigenvalueE=En+ ̄hω.
Nextweusethefactthattheharmonicoscillator,likethehydrogenatom,hasa
lowestenergystate(or”ground”state). Thattheremustbealowestenergystateis
clearfromtheuncertaintyprinciple,butwecanalsoseeitfromthefactthat
<p^2 > = <ψ|p^2 |ψ>=<pψ|pψ> ≥ 0
<x^2 > = <ψ|x^2 |ψ>=<xψ|xψ> ≥ 0 (9.37)
wherewehaveusedthehermiticityofxandp,andalsothefactthatthenormof
anynon-zerovectorisgreaterthanzero. Then
<H>=
1
2 m
<p^2 >+
1
2
k<x^2 > ≥ 0 (9.38)
This proves that a groundstate exists, withan energy greaterthan or equal to
zero. However,theexistenceofagroundstateseemstocontradictthefactthat,by
operatingonthegroundstatewiththeloweringoperatora,wegetastatewithstill
lowerenergy.Denotethegroundstatebyφ 0 ,theground-stateenergybyE 0 ,andlet
φ′=aφ 0 .Then,accordingtoeq. (9.32)
H ̃(aφ 0 )=(E 0 − ̄hω)(aφ 0 ) (9.39)
Thisequationissatisfiedifφ′=aφ 0 isaneigenstateofH ̃withenergyE 0 − ̄hω,which
meansthatφ 0 isnotthegroundstate,orif
aφ 0 = 0 (9.40)
Sincewehaveprovedthat agroundstateexists, thentheloweringoperatormust
”annihilate”thegroundstate;i.e. bringitto0,asshownin(9.40).Equation(9.40)
isafirst-orderdifferentialequationforthegroundstate:
(
√
mωx+ ̄h
1
√
mω
∂
∂x
)
φ 0 = 0 (9.41)