QMGreensite_merged

8.4. THEFINITESQUAREWELL:BOUNDSTATES 133

EvenParityBoundStates

Therequirementthatφ(x)=φ(−x)alsoimpliesthatA=B. Sowehavealto-
gether

φI(x) = Ae−

√

2 mEx/ ̄h

φII(x) = CEcos

[√

2 m(V 0 −E)x/ ̄h

]

φIII(x) = Ae

√ 2 mEx/ ̄h

(8.55)

Thecontinuityofthewavefunctionatx=±arequires

Ae−

√

2 mEa/ ̄h=C

Ecos

[√

2 m(V 0 −E)a/ ̄h

]

(8.56)

whilethecontinuityofthefirstderivativeatx=±agives

√

2 mEAe−

√

2 mEa/ ̄h=

√

2 m(V 0 −E)CEsin

[√

2 m(V 0 −E)a/ ̄h

]

(8.57)

Dividingequation(8.57)byequation(8.56)resultsinatranscendentalequation
forE=−E
√
E=

√

V 0 −Etan

[√

2 m(V 0 −E)a/ ̄h

]

(8.58)

whichdeterminestheenergyeigenvalues.
Thistranscendentalequationcanbesolvedgraphically,byplottingtheleftand
righthandsidesoftheequationtofindthepointsatwhichthetwocurvesintersect.
TheresultisshowninFig.[8.15].Welearnfromthefigurethatthesolutions,denoted
byEn,withnanoddinteger,areallatE<V 0. ThisisbecauseforE>V 0 thetangent
ineq. (8.58)becomesahyperbolictangent,sotherhsdiverges to−∞asx→∞,
neveragainintersecting

√

E. EachsolutionEnislocatedbetweenpointswherethe
tangentequals0;these”nodes”occurat

E=V 0 −(kπ)^2

̄h^2

2 ma^2

(k= 0 , 1 , 2 ,...) (8.59)

andsinceE>0,thismeansthatthenumberKofeven-parityenergyeigenvaluesis
thelargestnumbersuchthat

[(K−1)π]^2

̄h^2

2 ma^2

≤V 0 (8.60)

Thusthereareafinitenumber-andnolessthanone!-ofeven-parityboundstates.
ThenumberKofsuchboundstatesincreasesasthewellbecomesdeeper(largerV 0 ),
andwider(largera).

OddParityBoundStates