QMGreensite_merged

270 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY

andthesecond-ordercorrectiontotheenergy,

E^2 n = 〈φ(0)n |V|φ^1 n〉

= 〈φ(0)n |V

∑

i(=n

〈φ(0)i |V|φ(0)n 〉

En(0)−E(0)i

|φ(0)i 〉

=

∑

i(=n

∣∣

∣〈φ(0)n |V|φ(0)i 〉

∣∣

∣

2

En(0)−E(0)i

(17.37)

You maywonderhowmuchof thisyou reallyhavetoremember. Theaverage

quantumphysicistcanrecitefromfrommemorytheresultforthewavefunctionto

firstorder,andtheenergytosecondorderinλ.Sothesearetheresultstoremember:

φn = φ(0)n +λ

∑

i$=n

〈φ

(0)

i |V|φ

(0)

n 〉

E

(0)

n −E

(0)

i

φ

(0)

i

En = En(0)+λ〈φ(0)n |V|φ(0)n 〉+λ^2

∑

i$=n

∣∣

∣∣〈φ(0)

n |V|φ

(0)

i 〉

∣∣

∣∣

2

E

(0)

n −E

(0)

i

(17.38)

17.1 Validity of Perturbation Theory

Perturbationtheoryworks whentheperturbing potentialV′ issmallcomparedto
H 0. But...whatdowemeanby“small”? AgoodruleofthumbisthatV′issmallif
thefirstordercorrectionλφ(1)n ismuchlessthanthezeroth-orderwavefunction,which
requires(atleast)that

λ|c^1 in| 51 =⇒ λ

∣∣

∣∣

∣∣

〈φ

(0)

i |V|φ

(0)

n 〉

En(0)−Ei(0)

∣∣

∣∣

∣∣^51 (17.39)

Inotherwords,the“matrixelement”Vin′ oftheperturbingpotentialismuchsmaller
thanthecorrespondingenergydifference

|Vin′|≡λ|〈φ

(0)

i |V|φ

(0)

n 〉|^5 |E

(0)

n −E

0

i| (17.40)

Thisis,ofcourse,fori+=n(otherwisetheenergy differenceistriviallyzero). For
i=n,werequirethatthefirst-ordercorrectiontotheenergyissmallcomparedto
thezeroth-orderenergy,i.e.

En(1) 5 E(0)n =⇒ |〈φ(0)n |V|φ(0)n 〉| 5 En(0) (17.41)