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282 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY

TherearethreevaluesofEtoconsider:

CaseI: E= 3 a 0

Matrixmultiplicationonthelefthandsideof(17.115)givesus

3 a 0







c

0

a

0





= 3 a 0







a

b

c

d





 (17.116)

fromwhichweconcludethat

c=a , b=d= 0 (17.117)

Normalizationof

φ%=







a

0

a

0





 (17.118)

givesusa= 1 /

√

2. Sotofirstorderinλ=eEzwehaveaneigenstate

|φ〉E2+=

1

√

2

[

|φ 200 〉+|φ 210 〉

]

(17.119)

forenergyeigenvalue

E2+=E 2 (0)+ 3 ea 0 Ez (17.120)

CaseII: E=− 3 a 0

3 a 0







c

0

a

0





=− 3 a 0







a

b

c

d





 (17.121)

andtherefore

c=−a , b=d= 0 (17.122)

Normalizationof

φ%=







a

0

−a

0





 (17.123)