Biomedical Engineering Reference
In-Depth Information
we now have to think about visualization. Two popular ways to visualize such objects are as
contour diagrams or as surface plots. The two representative Figures 11.5 and 11.6 refer to
the solution
n
=
=
2, and in keeping with the importance of the Born interpretation,
I have plotted the squares of the wavefunctions. The horizontal axis is
x
, and the vertical
axis is
y
.
4 and
k
Figure 11.6
Two-dimensional box, surface plot
11.6 Three-Dimensional Infinite Well
It should be obvious by now howwe proceed to three dimensions. The relevant Schrödinger
equation is
∂
2
ψ
∂
x
2
∂
2
ψ
∂
y
2
∂
2
ψ
∂
z
2
8π
2
m
(ε
−
U
0
)
+
+
+
ψ
=
0
(11.14)
h
2
which can be solved by the standard techniques discussed above. Application of the bound-
ary conditions gives three quantum numbers that I will write
n
,
k
and
l
. The solutions
are
n
2
h
2
8
mL
2
+
k
2
h
2
8
mL
2
+
l
2
h
2
8
mL
2
ε
n
,
k
,
l
=
U
0
+
2
L
3/2
sin
k
π
y
L
sin
l
π
y
L
sin
n
π
x
L
(11.15)
ψ
n
,
k
,
l
=
n
,
k
,
l
=
1, 2, 3, ...
