New trends in physics teaching, v.4; The ... - unesdoc - Unesco

New Trends in Physics Teaching IV
to deal simply and effectively with statistical arguments - this involves choosing the best systems
to consider, the kind of statistics to emphasize and the mathematical resources to use; (4) how to
trade rigour against intuitive understanding (and working out what the latter means in this context);
and (5) just what physical systems, or types of systems, to emphasize (engines, electrochemical
cells, vapours etc.).
Feynman shows that he is aware of many of these problems: ‘it is a difficult subject and the
best way to learn it is to do it slowly. The first thing is to get some idea, more or less, of what
ought to happen in different circumstances, and then later. . . we wil formulate it better.’ On the
macroscopic - microscopic dilemma: ‘The deepest understanding. . . comes. . . from understanding
the actual machinery underneath, and that is what we shall do: we shall take the atomic
viewpoint from the beginning, and use it to understand the properties of matter and the laws of
thermodynamics.’ On the classical/quantum dilemma (what statistics to use) ‘. . . many things
that we wil deduce by classical physics wil be fundamentally incorrect. . . however, we shall
indicate in every case when a result is incorrect.’
Feynman offers an important insight into this last dilemma when he writes: ‘It turns out that. . .
although most problems are more difficult in quantum mechanics than in classical mechanics,
problems in statistical mechanics are much easier in quantum theory.’ The reason is that in
quantum theory one can count: states become discrete instead of continuous, so that integrals
turn into addition sums.
Feynman’s decisions about many dilemmas are clear. He firmly emphasizes microscopic interpretations.
His statistical arguments are as simple as they can be made: no permutations or
combinations are used at all. All the arguments concern Maxwell-Boltzmann statistics, though
he takes opportunities to demonstrate the failure of classical statistics. He chooses a clear focus
for argument in the Boltzmann factor, a choice enabling him to illustrate the wide variety of
applications for the ideas.
A remarkable and novel chapter on the ratchet and pawl as a heat engine attempts to link the
otherwise quite distinct macroscopic and microscopic sets of arguments. The key to understanding
the nature of his approach is its selectivity. Figure 1 shows just how selective it is, as it
indicates how few of the topics discussed by others are picked out by Feynman.
Choices amongst statistical arguments
Any decision to offer a statistical approach hinges on finding some simple treatment of the
statistics. Feynman finesses the problem. Rather than do statistics, he treats a clearly statistical
problem in a non-statistical style, choosing to look at the exponential distribution with height
of the density of an isothermal gas, from which he extracts the Boltzmann factor.
The exponential atmosphere
Feynman’s neat trick is to produce the simple argument that in an isothermal atmosphere in
equilibrium in a gravitational field, the density, and so the ratio of the numbers of molecules
per unit volume at different heights, is given by
n/n, = exp(-mgh/kT)
He argues that there must be a difference of pressure across any small interval dh, just sufficient
to support the weight of the gas in that interval, so that
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