Biomedical Engineering Reference
In-Depth Information
atoms at 300 K, then each atom will have
2
k
B
T
kinetic energy, according to the equiparti-
tion of energy principle. A short calculation suggests that we will find typically quantum
numbers as high as 10
9
; energy is certainly quantized, but the separation between the energy
levels is minute in comparison to
k
B
T
. Under these circumstances, it is possible to treat
the energy levels and the quantum numbers as if they were continuous rather than discrete.
This is called the
continuum approximation
.
Direct calculation shows how the number of quantum states lying in a given small interval
increases with energy (Table 12.1).
Table 12.1
Properties of quantum states. Here E
=
h
2
/
8
mL
2
Number of quantum states with energy lying
between
(100 and 110)
E
85
(1000 and 1010)
E
246
(10 000 and 10 010)
E
1029
(100 000 and 100 010)
E
2925
(1 000 000 and 1 000 010)
E
8820
What matters in many experiments is not the precise details of an energy level diagram,
rather a quantity called the
density of states D
(ε) defined such that the number of states
between ε and ε
dε is
D
(ε)dε.
Figure 12.1 shows a plot of the number of states versus the square root of the energy,
and it seems that there is a very good linear relationship.
+
1
·
10
4
5000
0
0
200 400
Square root of energy
600
800
1000
Figure 12.1
Square root relationship
I want now to establish that this is indeed the general case, and I can do this using a
process called
Rayleigh counting
.
