Biomedical Engineering Reference
In-Depth Information
The (time-independent) Schrödinger equation is
∂
2
ψ
∂
x
2
∂
2
ψ
∂
y
2
8π
2
m
(ε
−
U
0
)
+
+
ψ
=
0
(11.12)
h
2
and thewavefunction nowdepends on
x
and
y
. Straightforward application of the 'separation
of variables' technique discussed above gives the following wavefunctions and energies:
n
2
h
2
8
mL
2
+
k
2
h
2
8
mL
2
ε
n
,
k
=
U
0
+
sin
k
π
y
L
L
sin
n
π
x
2
(11.13)
ψ
n
,
k
=
L
n
,
k
=
1, 2, 3, ...
There are now two quantum numbers that I have decided to call
n
and
k
. Each quantum
number can equal a positive integer. The wavefunction is a product of sine functions, and
0.444
0.444
0.444
3.556
3.111
2.667
2.222
1.778
1.333
0.889
60
3.556
3.556
3.556
3.111
1.778
2.222
0.444
0.889
2.222
0.444
0.889
0.444
0.889
1.333
3.111
3.111
2.667
2.667
2.222
2.667
1.333
1.778
1.778
1.333
0.444
40
0.444
0.444
0.444
3.556
3.111
2.667
1.778
0.889
0.444
20
3.556
3.556
1.778
2.222
3.556
3.111
2.222
3.111
3.111
0.444
0.889
1.333
2.667
0.444
0.889
1.333
2.667
2.667
2.222
0.444
0.889
2.222
1.778
1.778
1.333
1.333
0
0
20
40
60
Figure 11.5
Square of wavefunction, contour view
