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Re: [Phys-l] Quantum of action

On 01/07/2012 09:22 AM, Moses Fayngold wrote:

How come then did Plank, Sommerfeld and others conclude that h was a
quantum of action?

It's complicated. Here's part of the story:

1) In the 1890s, long before there was any quantization, long before
Planck, Einstein, or Sommerfeld got involved, Boltzmann had shown
you could calculate the entropy of a free particle by distributing
the probability over "cells" in phase space. To do this you need
some scale factor i.e. some *unit of area* for measuring area in
phase space; otherwise the enterprise fails all sorts of tests
including scaling analysis aka dimensional analysis.

Without loss of generality we can use the symbol "h" to denote
this unit of area in phase space. Note that areas in phase space
have the same dimensions as action.

In classical (pre-1900) thermodynamics, the value of h is arbitrary.
Changing the value of h shifts the zero of entropy, which has no
effect on any classical observables.

2) In October 1900, Planck presented a paper
"Ueber eine Verbesserung der Wienschen Spectralgleichung"
(i.e. "On an improvement of Wien's spectral equation")

in which he presented the equation by fiat, and justified it
a_posteriori by saying it agreed with experiment.

The equation contains one adjustable parameter, which is well
determined by fitting to the known black-body spectrum.

3) It is a rule of cognitive science in general (and pedagogy in
particular) that when you come across a new idea, you should
check how it is related to other things you know. Initially,
the Planck radiation formula was not related to anything else
(except experiment). It was little more than numerology. This
must have bugged Planck immensely.

Later in 1900, Planck found a way to connect the radiation formula
to entropy.
"Ueber das Gesetz der Energieverteilung im Normalspectrum"
(i.e. "On the Law of Energy Distribution in the Normal Spectrum")

This is a Big Deal, because it tells us something about thermo
that we never knew before, namely the value of h ... and hence,
the zero of entropy. Entropy is no longer gauge-invariant.

So ..... Planck's first mention of Boltzmann, entropy, and/or energy
quantization is not his first mention of the radiation formula. I
don't know how he "originally" derived the formula; he may have used
a Ouija Board for all I know.

The /second/ time he derived the formula, he used quantization and
entropy arguments. He cited Boltzmann. At this point, the connection
to phase space was baked in.

Originally, Plank, Einstein, & others were talking about energy
quantization Delta E = h omega, with h already figuring as a
universal constant.

That's not the "original" story; see above.

It's not even the correct physics. In quantum mechanics and in
thermodynamics, the states are STATES; they are not "energy
states". To say (almost) the same thing another way, the action
is not quantized. Phase space is not quantized, not in units of
h or in any other units. Area divided by h tells you how many
states there are in the *basis* set ... but the basis states are
not the only states. This can be understood in terms of vector
analysis at the high-school level: You can take linear combinations
of the basis vectors to get as many vectors as you want. The
quantum mechanical Hilbert space is just a slightly fancy vector
space, and the same principles apply.

As a simple example: Consider an electrical LC oscillator. If
you measure the energy, the energy is quantized ... but if you
measure the voltage, the voltage is not quantized. If you try
to infer the energy (expectation value) from the voltage, the
inferred energy is not quantized. Let's be clear: energy
eigenstates are not the only states. They're not even the only
way of choosing basis states.

Pulsed-NMR experiments are another familiar example. It is
very common to flip the system /halfway/ from one energy level
to another.

Also, is there a way to derive quantization of action within the
framework of the new quantum theory

No. See above.

the Bohr-Sommerfeld condition only proves the
quantization of the quantity S + Int H dt.

That's a distinction without a difference, if we assume that
energy is conserved i.e. that the Hamiltonian is time-invariant.

========================

I would hasten to add that from the pedagogical point of view, the
history doesn't matter. Students, especially in the introductory
course, should be given the best available explanation, not the
most ancient explanation.

Generally speaking, old folks are interested in history, whereas
young folks are not.

Physicists, by and large, are lousy historians.

I'm not a historian. I probably can't tell the difference between
good history and mediocre history ... but I can recognize lousy
history when I see it.

When I see a historical statement in a physics text, I start by
assuming it is some combination of bogus history, bad pedagogy,
and wrong physics. For example, I was flipping through Serway &
Faugnn the other day, The section on mass and energy starts out
with the statement
"Einstein's E_R = mc^2 is one of the most famous equations of
the 20th century."

I'm glad to see the subscript "R" there, because that helps convey
the correct physics ... but putting it there falsifies the history.
The *famous* version of the equation is simply E = mc^2 ... so (as
is so often the case) there is a tradeoff involving some combination
of bogus history, bad pedagogy, and/or wrong physics. There are a
lot of people on this list who think history is important. I say
if the history is worth doing, it's worth doing right. I don't see
where textbook authors get a license to falsify the history for their
own convenience.

The next paragraph says
"In 1905, Einstein derived a new equation for kinetic energy based
on the principles of special relativity:

mc^2
KE = ----------------- - mc^2 [2]
sqrt(1-v^2/c^2)


This is bogus history *and* bad pedagogy.
a) I've read Einstein's 1905 papers, and unless I am overlooking
something, he never said that.
b) Rather than deriving or even explaining what is going on, the
authors are effectively telling the students: "Believe this equation
because we say so." Actually they are not upfront enough to even
say that; instead the message is "Believe this equation because
we say Einstein said so."

They go on to use equation [2] to (sorta) derive the E_R = mc^2
equation, which makes it look like they are being "scientific" ...
but it's a sham, because equation [2] was pulled out of thin air,
not connected to anything else ... and therefore it would be just
as logical to pull the E_R = mc^2 equation out of thin air. Thus
we have the appearance of logic without any real logic ... just a
thinly-disguised appeal to authority.

Some teachers wonder why the students don't exhibit much in the
way of critical thinking. Gaaack! I can tell you why. They've
been taught, over and over again, that in school, critical thinking
will just get you into trouble.

It is sometimes argued that the history is important, because
it provides a lesson about the /process/ of doing science. Alas,
the textbook version of history is so oversimplified as to grossly
understate how hard it is to do real science. As such it is a
disservice to students, and an insult to all scientists -- past,
present, and future.

If people on this list want to learn the history, I encourage that.
Specifically, I encourage people to learn the *real* history. Read
Kuhn's book, and then go read some of the old-time papers. You will
discover that the old-time scientists were amazingly clever about
some things, and also amazingly confused about some things.

I beseech you to keep the history out of the introductory course.
The intro course should keep things simple, but the real history
is not simple. Delving into the real history would be utterly
impossible due to time constraints, and would be bad pedagogy
even if you had unlimited time. If you're not going to teach the
real history, teaching false history is worse than nothing.

There is no law that says pedagogy must recapitulate phylogeny.