3.3 Solutions
175
d
2
ψ
dx 2 +
2m E
2
ψ = 0
d
2
ψ
dx 2 + α
2
ψ = 0
( 6 )
with α
2
=
2m E
2
(7)
ψ 2 = C sin
Odd
αx + D cos
even
αx
(8)
In this region either odd function must belong to a given value E or even
function, but not both,
Region 3; (E < V 0 )
Solution will be identical to (4)
ψ 3 = Ae
βx
+ Be
−βx
But physically accepted solution will be
ψ 3 = Be
−βx
(9)
because we must put A = 0 in this region where x takes positive values if the
wave function has to remain finite.
Class I (C = 0)
ψ 2 = D cos αx
(10)
Boundary conditions
ψ 2 (a) = ψ 3 (a)
(11)
dψ 2 /dx| x=a = dψ 3 /dx| x=a
(11a)
These lead to
D cos (αa) = B e
−βa
(12)
− D α sin(αa) = −B β e
−βa
(13)
Dividing (13) by (12)
α tan αa = β
(14)
Class II (D = 0)
ψ 2 = C sin(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
C sin(−αa) = −C sin(αa) = Ae
+βa
(18)
Cα cos(αa) = Aβe
βa
(19)
Dividing (19) by (18)
α cot (αa) = −β
(20)