9.1. RAISINGANDLOWERINGOPERATORS 147
whichhastheuniquesolution
φ 0 =Ne−mωx(^2) /2 ̄h
(9.42)
TheconstantN isdeterminedfromthenormalizationcondition
1 = N^2
∫∞
−∞
dxe−mωx
(^2) / ̄h
= N^2
√
π ̄h
mω(9.43)
sotheground-stateeigenfunctionis
φ 0 =[
mω
π ̄h] 1 / 4
e−mωx(^2) /2 ̄h
(9.44)
Thecorrespondingground-stateenergyE 0 is
H ̃φ 0 = h ̄ω(a†a+^1
2
)φ 0
1
2
̄hωφ 0=⇒ E 0 =1
2
̄hω (9.45)Thestatewiththenext-to-lowestenergyisobtainedbyoperatingwiththeraising
operatora†onthegroundstate:
φ 1 =c 1 a†φ 0 (9.46)wherec 1 isanormalizationconstant. Sincetheraisingoperatorraisestheenergyby
̄hω,theenergyofthisstateis
E 1 = E 0 + ̄hω= ̄hω(1+1
2
) (9.47)
Thestatewiththenexthigherenergyis
φ 2 ∝ a†φ 1
= c 2 (a†)^2 φ 0 (9.48)withenergy
E 2 = E 1 + ̄hω= ̄hω(2+