Chapter 17
Time-Independent Perturbation
Theory
ConsideraHamiltonianwhichhasthisform:
H=H 0 +alittlebitextrapotential (17.1)
whereH 0 isaHamiltonianwhoseeigenvalueequationisalreadysolved:
H 0 φ^0 n=E^0 nφ^0 n (17.2)
Itthenmakessensethatthesolutionstotheeigenvalueequation
Hφn=Enφn (17.3)
canbewritten,foreachn,as
φn = φ^0 n+alittlebitextrafunction
En = En^0 +alittlebitextraconstant (17.4)
Anexample:
H=−
̄h^2
2 m
d
dx^2
+
1
2
kx^2 +λx^4 (17.5)
Inthiscase
H 0 =−
̄h^2
2 m
d
dx^2
+
1
2
kx^2 (17.6)
istheHamiltonianofaharmonicoscillator,whoseenergyeigenvaluesandeigenfunc-
tionsarewellknown,and
alittlebitextrapotential=λx^4 (17.7)
AnotherexampleisthecaseofaaHydrogenatominanexternalelectricfield,directed
(say)alongthez-axis
H=−
h ̄^2
2 m
∇^2 −
e^2
r
+eEz (17.8)