Re: Uncertainty principle, fourier transform

mmeron@xxxxxxxxxxxxxxxxxx wrote:

We should start at the 17th century. This was the century of Newton,
but also of couple other luminaries who figure in the story, namely
Fermat and Huygens. Fermat dabbled in optics, among other things, and
established the principle that light propagates among any two points
along the shortest (later generalized to "stationary") optical path
consistent with the constraints of the system. Note that he didn't
take a stand on whether light is in fact a wave or a stream of
particles, just established practical rules along the lines of "that's
how light will propagate in a given situation". Huygens believed that
light consists of waves and created a general model of wave
propagation (model applicable to any wave, not just light, so its
standing didn't depend on how the argument about light will turn out).
In this model, any point the wavefront passes through is a secondary
source, radiationg in "all" directions, and the specific pattern we
observe is obtained through a superposition of radiation from all
those sources, with the miracle of interference taking care of
details. Huygens' model was still semi-qualitative because all the
math required wasn't there yet, but all the key ingredients were
there. Note that, in this model, given any starting and ending point
the wave propagates along all the possible paths joining these points.
The third of the bunch, Newton, believed that light is made of
particles but (same as the others) didn't conclusively find out one
way or the other and, as far as quantitative work is involved, left
his main imprint on the dynamics of particles.

OK, fast forward some 100-150 years. We've some developments. On one
hand, Newtonian mechanics has been greatly advanced, through the work
of Maupertois, D'Alembert, Lagrange, Poisson, Hamilton (I'm sure there
are more names that can be added) and took the mature form of the
"Least Path Principle". By interesting coincidence this looks quite
similar to Fermat's principle in optics. There are differences, of
course as the mechanical "path" lies in configuration space, not just
the plain regular space of Fermat, and there is some other quantity
called "action", the integral of which is being minimized. Still the
principle is similar. In the meantime, there are also advances in
optics and finally, through the work of Young and Fresnel (and, again,
many others) it is established that light is indeed a wave. Also,
we've Kirchoff who puts the business of waves propagating along all
possible paths on a solid mathematical footing, in the form of a
theorem which can be used for quantitative calculations.

OK, so here we've something which at first look may appear as a puzzle
(to put it midly). On one hand we've solid evidence that light is a
wave and we've the math describing its propagation along all possible
paths. On the other hand, we've Fermat's rule where light propagates
along a single path only, the stationary path. There is an apparent
contradiction here. Can it be reconciled? Turns out that yes, it
can, no problem. The thing is, yes, a wave propagates from point A to
point B along all possible paths but these various contributions
arrive at B with different phases. The phase difference, in general,
equals delta_pathlength/wavelength. So, if the wavelength is very
short, even small differences in path length give rise to large phase
differences and the various components interfere destructively,
leaving little if anything. The really significant contributions are
obtained only from these paths where the length difference between one
path and its neighboring paths is, to first order at least, zero.
Now, this is just the definition of "stationary path". So we get the
important result that starting from propagation along all paths and
taking the limit of wavelength going to zero, the only thing left are
contributions from stationary paths. In optics this is referred to as
"ray optics obtains from ray optics at the short wavelength limit".
And what happens if we refrain from taking the limit? Why, then we
have wave optics which is a perfectly legitimate form, in fact the
more general form (and highly useful one).

So, at this point (said point being somewhere in the middle of the
19th century) we can in principle say "hey, this worked with optics,
why not with mechanics?" We can start thinking about some more
general "wave mechanics" such that the mechanics we've, with its least
action principle, is the "ray mechanics", obtainable from the more
general one in the short wavelength limit. Only, we've nothing to
proceed with. Neither an evidence of any sort of "waviness" in
mechanics, nor any meaningful way to assign any sort of "wavelength"
to mechanical systems. Mathematical similarity is not enough. Any
mathematical model can always be represented as an appropriate limit
of some yet more general model, this doesn't mean that such
"embedding" carries any physical meaning. So that's where matters
stay till the beginning af the 20th century where we've new
developments. On the one hand we bagin to acquire evidence of some
sort of "machanical waviness". On the other hand, we have now a way
to meaningfully relate mechanical action to wavelength, courtesy of
the newly discovered Planck constant. And there is a natural "limit
taking" criterion since the wavelengths thus assigned are proportional
to the Planck constant. So, taking the zero wavelength limit is just
equaivalne to saying "if h would've been zero, we would have purely
classical mechanics".

To make a long story short, we're now in a position to repeat what was
done for optics, i.e. create a more general "wave mechanics" such that
the mechanics we had before is its corresponding "ray mechanics". And
that's step one in obtaining quantum mechanics, just get the wave
formulation, then "refrain from taking the limit". So far, we're
really not doing anything earth shaking, just repeating what was
already done in optics (Timo, Andy, I'm sure you'll agree with this).
Only, this is not the whole story. One more step is needed and this
one has no previous "classical precedent".

The thing is, when we change from "rays" to "waves" we replace stricly
localized entities with "diffuse entities". A ray starts at point A
and arrives at B. A wave arrives "a bit here, a bit there, a bit
everywhere". And we've no problem with this since that's what waves
are supposed to do and (more important) that's what we actually
observe them to do.

With particles, it is a different story. Not only we don't expect
them to be "spread around" (that wouldn't have been a problem because
we're already in the mode of adjusting expectations to observations,
not the other way around) but, when observing one particle at a time,
we don't see them spread around. No, any particle is detected as a
localized event. Yet, the localization of this event varies from one
particle to the next, with the envelope of all events being actually
spread around. So, the big issue becomes "what the hell is waving
here?"

With classical waves (including wave optics) the answers are clear.
What is waving is a physically obesrvale parameter. Electric (or
magnetic) field in the case of light waves, pressure with sound waves,
displacement with elastic waves in solids etc. Note that all these
physical quantities are non-localized fields. But what "waves" in
the QM case is not anything like this. At most you can say that in
the limit of a large number of repeated measurements the wave tells
you something about their distribution. So, you're forced to accept
the idea that what this wave does is to determine probabilities. This
is the second step on the way to QM and it has *no previous
precedent*.

As for the last tidbit, of how exactly is the wave related to the
probability, well lets take our guide from optics (again). In the
macroscopic case, the amount of light delivered to any particular
spot (where by the "amount of light" we may mean energy, momentum
whatever) is proportional to the absolute squared amplitude of the
wave at said spot. On the other hand, if we take the microscopic
picture and talk in terms of photons (it is late 1920s now, so we can
talk about them), the amount delivered is proportional to the number
of photons which (assuming we already acepted the probability
interpretation of the wave function) is proportional to the
probability as determined by said wave. What follows from
this is that it is natural to to assume that said probability is
simply proportional to the absolute squared amplitude of the wave.
Note this is not a "proof" but it is reasonably convincing.

So, to answer your question above, after this long detour, equating
the square of the amplitude with intensity was a standard optics
approach. Equating this, in turn, with probability, was hardly
standard anything. But, if anybody could tie it together, a good
optics guy could. And Max Born, who was the head of the physics
department in Goetingen at the time, Heisenberg's thesis advisor (as I
recall), was an optics guru.

Wow. Thank you for that masterful exposition: how far along is your
new book on the history of modern science? :-)

I have nothing to add, except that you motivate me to hook up my
printer again, and even do some more background reading in these areas.
Now, if we could just enlist one or two more "graduate students", and
encourage some of your colleagues (whom I see you left some bait for in
that passage, though they've not yet risen to take it) to engage more
substantively, and there might yet be life in the old monster: is that
a galvanic twitch I see in the left ankle?

Oh yeah... there is something I don't get even on this qualitative
level: given that light (say) is propagating along all possible paths,
but that paths differing substantially from the least action path are
distributed in phase so as to cancel (do I have that right?), why don't
we see a contribution from paths far enough out so that light arriving
at a point is again in phase with direct propagation? Does this
question make sense? Or am I conflating different approaches?

Oh... in fairness... I think we had got so far as "all is the same,
except measurement" last time -- though I but vaguely recollected it.

.

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