QMGreensite_merged

8.4. THEFINITESQUAREWELL:BOUNDSTATES 131

BoundStates E< 0

Webeginwithadiscussionoftheboundstates.InRegionsIandIIItheSchrodinger
equationisthesameasafreeparticleequation,andthesolutionis

φI(x) = a 1 ei

√

2 mEx/ ̄h+a 2 e−i

√

2 mEx/ ̄h

φIII(x) = b 1 ei

√

2 mEx/ ̄h+b

2 e

−i

√

2 mEx/ ̄h (8.39)

However,sinceE<0,thesquare-rootisimaginary.WritingE=−E,wehave

φI(x) = a 1 e−

√

2 mEx/ ̄h+a

2 e

√

2 mEx/ ̄h

φIII(x) = b 1 e−

√ 2 mEx/ ̄h

+b 2 e

√ 2 mEx/ ̄h

(8.40)

Ifψisaphysicalstate,wemustseta 2 =b 1 =0;otherwisethewavefunctionblows
upexponentiallyasx→±∞andthestateisnon-normalizable.Then

φI(x) = Ae−

√

2 mEx/ ̄h

φIII(x) = Be

√

2 mEx/ ̄h (8.41)

NowsupposewechooseAtobearealnumber.Thenφ(x)mustberealeverywhere.
ThereasonisthattheSchrodingerequationisalinearequationwithrealcoefficients,
sowriting
φ(x)=φR(x)+iφI(x) (8.42)

itseasytoseethatφRandφImustbothsatisfytheSchrodingerequation

H ̃φR=EφR H ̃φI=EφI. (8.43)

Setting Ato beareal number meansthat φI = 0 forx> a, which meansitis
zeroeverywhere, because the Schrodingerequation saysthat φ′′(x) ∝ φ(x). The
conclusionisthatφ(x)isrealeverywhere;inparticular,theconstantBisalsoareal
number.
Letφ(x)beasolutionofthetime-independentSchrodingerequationwithenergy
En. Wecanshowthat
φ(x)=φ(−x) (8.44)

isalsoaneigenstateofH ̃,withthesame energy. Startwiththetime-independent
Schrodingerequation

−

̄h^2

2 m

d^2

dx^2

φ(x)+V(x)φ(x)=Eφ(x) (8.45)

makethechangeofvariablesx→−x

−

̄h^2

2 m

d^2

dx^2

φ(−x)+V(−x)φ(−x) = Eφ(−x)

−

̄h^2

2 m

d^2

dx^2

φ(x)+V(x)φ(x) = Eφ(x) (8.46)