17.2. EXAMPLE-THEANHARMONICOSCILLATOR 271
17.2 Example - The Anharmonic Oscillator
ConsidertheHamiltonian
H = −̄h^2
2 md
dx^2+
1
2
kx^2 +λx^4= H 0 +λx^4 (17.42)Theshiftintheenergy,tofirstorderinλ,is
∆En = λEn(1)
= λ〈n|x^4 |n〉 (17.43)whereweintroducethenotation,fortheharmonicoscillator,that
|n〉≡|φ(0)n > (17.44)Asusualinharmonicoscillatorproblems,ithelpstoexpressthepositionoperatorx
intermsofraisingandloweringoperators
x =√
̄h
2 mω(a+a†)x^4 =(
̄h
2 mω) 2
(a+a†)^4=
(
̄h
2 mω) (^2) [
a^2 (a†)^2 +aa†aa†+a(a†)^2 a+
+a†a^2 a†+a†aa†a+(a†)^2 a^2
]
+non-contributingterms (17.45)
Usingtheraising/loweringoperatorproperties
a†|n〉 =
√
n+ 1 |n+ 1 〉
a|n〉 =√
n|n− 1 〉 (17.46)wefind,e.g.,that
〈n|a^2 (a†)^2 |n〉 =√
n+ 1 〈n|a^2 a†|n+ 1 〉
=√
n+ 1√
n+ 2 〈n|a^2 |n+ 2 〉
=√
n+1(n+2)〈n|a|n+ 1 〉
= (n+1)(n+2) (17.47)Evaluatingalloftherelevanttermsinthisway,wefind
〈n|x^4 |n〉 =
(
̄h
2 mω) (^2) [
(n+2)(n+1)+(n+1)^2 +n(n+1)+n(n+1)+n^2 +n(n−1)
]
= 3
(
̄h
2 mω) 2
[1+ 2 n(n+1)] (17.48)