QMGreensite_merged

17.4. DEGENERATEPERTURBATIONTHEORY 279

tofindtworoots

E±=±

1

2 β^2

(17.96)

correspondingtoeigenstates

φ+ = aφ 10 +bφ 01

φ− = cφ 10 +dφ 01 (17.97)

Wefindtheseeigenstatesbysolving

1

2 β^2

[

0 1

1 0

][

a

b

]

=

1

2 β^2

[

a

b

]

1

2 β^2

[

0 1

1 0

][

c

d

]

= −

1

2 β^2

[

c

d

]

(17.98)

subjecttotheconditionthat

〈φ+|φ+〉 = a^2 +b^2 = 1

〈φ−|φ−〉 = c^2 +d^2 = 1 (17.99)

Thesolutionsare

φ+ =

1

√

2

[

1

1

]

=

1

√

2

[φ 10 +φ 01 ]

φ− =

1

√

2

[

1

− 1

]

=

1

√

2

[φ 10 −φ 01 ] (17.100)

Insteadoftwodegenerateenergies, E 10 =E 01 ,we nowhaveenergyeigenvaluesto
firstorderinλ

Hφ+ = E+φ+

Hφ− = E−φ− (17.101)

whereE=E(0)+λE,i.e.

E+ = E 10 +

λ

2 β^2

E− = E 10 −

λ

2 β^2

(17.102)

Wesaythatthedegeneracyis“liftedatfirstorder”bytheperturbation.