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