Biomedical Engineering Reference
In-Depth Information
0.8
0.8
1.599
2.399
1.599
3.198
2.399
3.198
3.998
1.599
0.8
2.399
0.8
1.599
0.8
0.8
0.8
1.599
2.399
1.599
3.198
3.998
2.399
3.198
0.8
1.599
2.399
0.8
1.599
0.8
Figure 12.10
(
Ψ
1,2
)
2
versus x
A
and x
B
The plots are of the square of the wavefunction (the 'out-of-plane' axis) versus the
x
coordinates of the two electrons along the in-plane axes. The diagrams imply quite different
probability densities for an arbitrary choice of the pair of coordinates; the probability
contours ought to be symmetrical about their diagonals. The problem can be removed if we
make use of the degeneracy mentioned above and construct a wavefunction that is either
the sum or the difference of the product functions with
x
A
and
x
B
interchanged. Each of
these two combinations gives a probability density that is unaffected by an interchange of
the names of the two particles. Allowing for the correct normalization, these combinations
are the following.
Symmetric:
1
2
Ψ
1,2
(
x
A
,
x
B
)
Ψ
2,1
(
x
A
,
x
B
)
Ψ
s
(
x
A
,
x
B
)
=
+
(12.19)
Antisymmetric:
1
2
Ψ
1,2
(
x
A
,
x
B
)
Ψ
2,1
(
x
A
,
x
B
)
Ψ
a
(
x
A
,
x
B
)
=
−
(12.20)
