QMGreensite_merged

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)

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