Biomedical Engineering Reference
In-Depth Information
12.2 Rayleigh Counting
First of all I plot out the quantum numbers
n
,
k
and
l
along the
x
,
y
and
z
Cartesian axes as
shown in Figure 12.2.
l
r
k
n
Figure 12.2
Rayleigh counting
For large quantum numbers the points will be very close together and there will be
very many possible combinations of the three particular quantum numbers
n
,
k
and
l
that
correspond to the same energy
l
2
h
2
8
mL
2
According to Pythagoras' theorem, each combination will lie on the surface of a sphere of
radius
r
where
ε
n
,
k
,
l
=
n
2
+
k
2
+
r
2
n
2
k
2
l
2
=
+
+
We therefore draw the positive octant of a sphere of radius
r
, as shown. The volume of the
octant also contains all other combinations of the three quantum numbers that have energy
less than or equal to the value in question. We only need consider this octant, since all
quantum numbers have to be positive. The number of states with energy less than or equal
to ε is
n
2
l
2
3/2
1
8
4π
3
+
k
2
+
or in terms of the energy
1
8
4π
3
8
mL
2
h
2
ε
3/2
d
r
, shown in Figure 12.3, which contains all
quantum states having energy less than or equal to ε
I now draw a second octant of radius
r
+
+
dε. The volume of the outer sphere is
1
8
4π
3
8
mL
2
h
2
+
dε)
3/2
(ε
