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+