3.3 Solutions
187
3.39 (a)
−
2
2m
∇
2
+ V
ψ(x, y, z) = Eψ(x, y, z)
( 1 )
Put V = 0
−
2
2m
∂
2
∂ x 2 +
∂
2
∂ y 2 +
∂
2
∂z 2
ψ(x, y, z) = Eψ(x, y, z)
Let
ψ (x, y, z) = X (x) Y (y) Z (x)
(2)
Y Z
∂
2 X
∂ x 2 + Z X
∂
2 Y
∂ y 2 + XY
∂ Z
∂z 2 = −
2m E
2
XY Z
Dividing throughout by XYZ
1
Y
∂
2 Y
∂ y 2 +
1
Z
∂
2 Z
∂z 2 = −
1
X
∂
2 X
∂ x 2 −
2m E
2
(3)
LHS is a function of y and z only while the RHS is a function of x only.
The only way (3) can be satisfied is that each side is equal to a constant, say –
α
2 .
1
X
d
2 X
dx 2 +
2m E
2 − α
2
= 0
∂
2 X
dx 2 +
2m E
2 − α
2
X = 0
Or
∂
2 X
∂ x 2 + β
2 X = 0
where
β
2
=
2m E
2
− α
2
(4)
X = A sin βx + B cos βx
Take the origin at the corner
Boundary condition: X = 0 when x = 0. This gives B = 0.
X = A sin βx
Further, X = 0 when x = a
Sinβa = 0 → βa = n x π
Or
β =
n x π
a
(n x = integer)
(5)