Biomedical Engineering Reference
In-Depth Information
text (Atkins 2006) says: 'No more than two electrons may occupy any orbital, and if two
do occupy it their spin directions must be opposite.'
In principle we must invoke the symmetry or antisymmetry of a wavefunction when deal-
ing with systems of two or more identical particles. For example, if we wish to describe two
ground-state hydrogen atoms, the total wavefunction must be antisymmetric to exchange
of the names of the two electrons, even if an infinite distance separates the two atoms.
Common sense suggests that this is not necessary, and the criterion is to ask by how much
the two wavefunctions overlap each other. If the overlap is appreciable, then we must worry
about the exclusion principle. If the overlap is negligible, we can disregard the symmetry
or otherwise of the total wavefunction.
12.10 Boltzmann's Counting Rule
We must now return to the three probabilities discussed in Section 12.5. Consider a simple
case of three particles A, B and C, each of which has a different energy.
Figure 12.14 shows three of the six possible arrangements of the three
distinguishable
particles, and, according to the Boltzmann counting procedure, we take account of all six
and they are all weighted equally when we calculate the number of ways
W
.
C
B
C
B
C
A
A
A
B
Figure 12.14
Boltzmann's counting method
This procedure is not consistent with the principles of quantum mechanics discussed
above. We certainly cannot label particles and make them distinguishable, and it is not
meaningful to enquire which particle is in one particular state. All we can do is to say how
many particles are in each state, and instead of six possibilities there is just one (with one
particle in each state).
We then have to consider whether the particles are fermions or bosons; if they are fermions
with spin
1
/
2
(like electrons), then each quantum state can hold no more than two particles,
one of and one of spin. If they are bosons, then the exclusion principle does not apply
and all the particles can crowd together in any one state, should they so wish. This is the
basis of the three different probabilities discussed in Section 12.5.
That leaves just one loose end: if all particles are either fermions or bosons, why does
the Boltzmann distribution work so well in explaining the properties of 'ordinary' gases?
