Biomedical Engineering Reference
In-Depth Information
Table 7.6
Arrangements and probabilities
A
B
C
W
=
no. of ways
Probability
7D
1D
1D
1
7
1
3
3/28
1
1
7
6
2
1
6
6/28
5
2
2
3
3/28
5
3
1
6
6/28
4
4
1
3
3/28
4
3
2
6
6/28
3
3
3
1
1/28
ways we can achieve the total energy with each individual distinguishable particle having a
certain amount of energy. For clarity, I have omitted the 'D' from all entries except the first.
The numbers in the fourth column are obviously related to factorials; if each particle has
a different energy then there are 3! ways in which we can allocate the energies amongst the
three particles. If two energies are the same then we have to divide the 3! by 2! and so on.
If we were to make a spot check of the quantum state occupancy, we would find on average
a situation where one particle had 7 D and the other two 1 D, for 3/28 of our measurements
(on the other hand, since this is a quantum system, I had better be a bit more precise and
say that if we were to prepare a very large number of such systems and measure the energy
distribution in each copy, we would find 7 D, 1 D,1Din3/28 of the systems).
Each arrangement is known as a
configuration.
Consider now amacroscopic systemcomprising
N
identical but distinguishable particles,
each of which has a similar quantum state diagram to that shown in Figure 7.5. The particles
are noninteracting. We can fix the total energy
U
and the number of particles, but we cannot
fix how the energy is distributed amongst the available energy levels (and it does not make
much sense to ask about the energy distribution amongst
N
individual particles, given that
N
might be 10
23
). The most sensible thing is to say that there are
N
1
particles with energy
ε
1
,
N
2
with energy ε
2
and so on. Suppose also that the highest possible quantum state
p
has
energy ε
p
.
Consideration of our simple example above shows that the number of ways in which we
can allocate the particles to the quantum states is
N
!
W
=
(7.6)
N
1
!
N
2
!
...
N
p
!
The fact that a particular quantum state may have no particles (or, in other words, it has a
zero probability) is accounted for by the fact that 0
1.
It turns out that for large
N
there is one particular configuration that dominates, in which
case the values of
N
i
are those that maximize
W
. At first sight they can be found by setting
the gradient of
W
to zero, but
N
i
cannot vary freely; they are constrained by the fact that
the number of particles and the total energy
U
are both constant
!=
N
=
N
1
+
N
2
+···+
N
p
U
=
N
1
ε
1
+
N
2
ε
2
+···+
N
p
ε
p
(7.7)
