QMGreensite_merged

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)