Biomedical Engineering Reference
In-Depth Information
Figure 11.7 summarizes the one-dimensional and the cubic three-dimensional infinite
well problems. The energies are plotted in units of
h
2
/8
mL
2
. The one-dimensional energy
levels are shown as the left-hand 'column' of the figure. They simply increase as
n
2
.
Energy - U
0
20
15
10
5
0
Figure 11.7
One-dimensional and cubic three-dimensional wells
The three-dimensional problem shows a much more interesting pattern. Firstly, many of
the energy levels are
degenerate
(that is to say, there are many
quantum states
with the same
energy). Some of the degeneracies are
natural
, for example ψ
1,2,3
, ψ
1,3,2
, ψ
2,1,3
, ψ
2,3,1
, ψ
3,1,2
and ψ
3,2,1
all have energy 14
h
2
/(8
mL
2
). There are also many
accidental
degeneracies; for
example there are 9 degenerate quantumstateswith energy 41
h
2
/(8
mL
2
) because 1
2
+
2
2
+
6
2
and 4
2
3
2
are both equal to 41. Secondly, as the energy increases, the energy levels
crowd together and the number of quantum states of near equal energy is seen to increase.
This crowding together does not happen in the one-dimensional case.
The square two-dimensional case lies somewhere in between, with fewer degeneracies.
The degeneracies do not occur if we take arbitrary sides for the two- and three-
dimensional regions. Nevertheless, their quantum states still crowd together as the quantum
numbers increase.
+
4
2
+
