1000 Solved Problems in Modern Physics

3.3 Solutions 175

d^2 ψ

dx^2

+

(

2 mE

^2

)

ψ= 0

d^2 ψ

dx^2

+α^2 ψ=0(6)

withα^2 =

2 mE

^2

(7)

ψ 2 =Csin

Odd

αx+Dcos

even

αx (8)

In this region either odd function must belong to a given valueEor even

function, but not both,

Region 3; (E<V 0 )

Solution will be identical to (4)

ψ 3 =Aeβx+Be−βx

But physically accepted solution will be

ψ 3 =Be−βx (9)

because we must putA=0 in this region wherextakes positive values if the

wave function has to remain finite.

Class I (C=0)

ψ 2 =Dcosαx (10)

Boundary conditions

ψ 2 (a)=ψ 3 (a) (11)

dψ 2 /dx|x=a=dψ 3 /dx|x=a (11a)

These lead to

Dcos (αa)=Be−βa (12)

−Dαsin(αa)=−Bβe−βa (13)

Dividing (13) by (12)

αtanαa=β (14)

Class II (D=0)

ψ 2 =Csin(αx) (15)

Boundary conditions:

ψ 2 (−a)=ψ 1 (−a) (16)

dψ 2 /dx|x=−a=dψ 1 /dx|x=−a (17)

These lead to

Csin(−αa)=−Csin(αa)=Ae+βa (18)

Cαcos(αa)=Aβeβa (19)

Dividing (19) by (18)

αcot (αa)=−β (20)