1000 Solved Problems in Modern Physics

(Grace) #1

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)
Free download pdf