Biomedical Engineering Reference
In-Depth Information
which is very similar to the Boltzmann factor. For sufficiently large electron energies, the
Fermi factor reduces to the Boltzmann factor. For electron energies below the Fermi level,
the factor can be written
exp
ε
ε
F
k
B
T
−
p
(ε)
≈
1
−
which simply tells us that nearly all the states are fully occupied.
We have therefore reached a point where three different types of particles in the same
three-dimensional infinite potential well show very different behaviour (Table 12.2).
Table 12.2
Density of states and average occupancies
Particle
Density of states
proportional to
Average occupancy
proportional to
exp
ε
k
B
T
'Classical' ideal gas
ε
1
/
2
−
1
exp
ε
k
B
T
ε
2
Photons in cavity
−
1
1
exp
(ε
−
ε
F
)
k
B
T
Electrons in metal
ε
1
/
2
+
1
12.6
Indistinguishability
Letme nowreturn to the infinite one-dimensional well, with just two noninteracting particles
present. I labelled the particles A and B, and showed in Chapter 11 that the solutions were
L
sin
n
A
π
x
A
sin
n
B
π
x
B
L
2
Ψ
n
A
,
n
B
=
L
E
n
A
,
n
B
=
n
A
+
n
B
h
2
8
mL
2
(12.17)
n
A
,
n
B
=
1, 2, 3, ...
Suppose one particle has a quantum number of 1, whilst the other has a quantum number
of 2. There are two possibilities
L
sin
1π
x
A
sin
2π
x
B
L
2
Ψ
1,2
=
L
(12.18)
L
sin
2π
x
A
sin
1π
x
B
L
2
Ψ
2,1
=
L
