QMGreensite_merged

134 CHAPTER8. RECTANGULARPOTENTIALS

Thistime, the requirement that φ(−x) = −φ(x)implies that B = −A, and
CE=0. Thewavefunctionhastheform

φI(x) = Ae−

√

2 mEx/ ̄h

φII(x) = COsin

[√

2 m(V 0 −E)x/ ̄h

]

φIII(x) = −Ae

√

2 mEx/ ̄h (8.61)

Fromcontinuityofthewavefunctionatx=±awehave

Ae−

√

2 mEa/ ̄h=COsin

[√

2 m(V 0 −E)a/ ̄h

]

(8.62)

andfromcontinuityofthefirstderivativeatx=±a

−

√

2 mEAe−

√

2 mEa/ ̄h=

√

2 m(V 0 −E)COcos

[√

2 m(V 0 −E)a/ ̄h

]

(8.63)

Dividingeq. (8.63)byeq. (8.62)givesusthetranscendentalequationfortheenergies
ofoddparityboundstates

√

E = −

√

V 0 −Ectn

[√

2 m(V 0 −E)a/ ̄h

]

=

√

V 0 −Etan

[√

2 m(V 0 −E)

a

h ̄

+

π

2

]

(8.64)

whichcanbesolvedgraphically, asshowninFig. [8.16]. Onceagain,thereareas
manyrootsastherearenodesofthetangent;thistimethenodesarelocatedat

E=V 0 −[(k+

1

2

)π]^2

̄h^2

2 ma^2

(k= 0 , 1 , 2 ,...) (8.65)

andthenumberofoddparitynodesisthelargestintegerMsuchthat

[(K−

1

2

)π]^2

̄h^2

2 ma^2

<V 0 (8.66)

Notethatfor

V 0 <

(

π

2

) 2

̄h^2

2 ma^2

(8.67)

therearenoodd-parityboundstates.

Tosumup,wehavefoundthatforafinitesquarewell

1. Thenumberofboundstateenergiesisfinite,andthereisatleastoneboundstate;


2. Thenumberofboundstatesincreaseswiththewidthanddepthofthewell;