From: Michael Lorrey (retroman@together.net)
Date: Sat May 30 1998 - 17:14:31 MDT
Daniel Fabulich wrote:
> On Sat, 30 May 1998, Christian Whitaker wrote:
>
> > The uncertainty principle states that it is impossible to precisely
> > measure a particle's momentum and position simultaneously. If one is
> > measured precisely, the other becomes unbounded. This has practical
> > significance towards the question of whether a photon is or is not blue
> > according to whether it lies within the range of 540-560 nm. As the
> > wavelength of the photon may lie arbitrarily close to the cutoff point
> > of 540 nm, the wavelength must be specified to infinite precision. The
> > wavelength of a photon is determined by the relationship x=h/E, where x
> > is the wavelength, h is Planck's constant, and E is energy. I will
> > further translate E into mc^2(which describes all aspects of momentum
> > but the direction). x=h/mc^2.
>
> <disclaimer> I do not know enough quantum to say very much about it;
> anything I do say may be wrong. Please be polite when you correct me.
> </disclaimer>
>
> However, I do know enough about light to say that the energy of light is
> NOT given by mc^2.
You are right. He is right. The energy is given by: e=h/x, or the Planck's
constant divided by the wavelength. Gven this energy measurement, you can derive
the mass value of a photon as a function of its energy, when that energy is
translated at the end of the photon's trajectory, i.e. when it is absorbed. Its
quantum 'mass' is given by m=h/xc^2. Because its energy can therefore be
measured in terms of mass, it can be described as a particle in certain
circumstances, i.e. when its kinetic energy is absorbed.
> This is the rest energy of particles which HAVE mass;
> photons don't. The REAL relativistic equation is actually E^2 = p^2*c^2 +
> m^2*c^4, where p here is the momentum, given by (g)^2*m^2*v^2 for normal
> particles, where (g) is given by 1/(1 - v^2/c^2)^0.5. Plug the equation
> for momentum into the relativistic equation and you get E^2 = (g)^2*m^2*
> v^2*c^2 + m^2*c^4. Factor this and you can get E^2 = (g)^2*m^2*c^2*(v^2 +
> c^2). For very small velocities, v^2 + c^2 ~= c^2 and (g) ~= 1. So we
> get E^2 = m^2*c^4, or E = m*c^2.
>
> Obviously, the approximations we make for the rest energy of slow moving
> particles is completely wrong; for light, which has no mass, the second
> term of the first equation drops off altogether, and we get E^2 = p^2*c^2,
> or E = p/c. The energy on a photon is given by h*f where f is the
> frequency, so we can get f = p/h/c. Using the definition of wavelength
> that x = c/f, we get x = h*c^2/p.
>
> We run up against a similar problem, however, when we do try to identify
> the frequency (though very different from the one which you had originally
> posited in your first post). h and c are indeed constants, so the more
> precisely you know a particle's wavelength the less precisely you know its
> position. Many people have taken the uncertainty principle to be saying
> that were we to know the precise location of a photon, it would exist
> EVERYWHERE; while this is often a useful way to think about it, this is
> not strictly correct. A photon whose wavelength is known perfectly may be
> said to exist *anywhere* in the entire universe; a photon whose position
> is known perfectly must be said to have *any* frequency.
>
> Does this contradict the idea of Tarskian truth statements? Not
> necessarily. Again, a large part of the reason who we have Heisenberg's
> uncertainty principle is because particles move about in an apparently
> random way, bounded by certain probability functions (which are themselves
> very specific and well-defined). So if we know the particle to exist at
> precisely one location, its momentum will be completely *random*. By
> definition, no equation can predict the completely random; we cannot
> therefore state, a priori, what that momentum will be.
>
> Does that mean that the blueness of a particle whose position is known
> perfectly is in an "ambiguous" state of blueness? No, it means that the
> particle is in a *random* state of blueness. If you want to find out
> perfectly if it's blue or not, you'll have to measure its momentum
> perfectly, at which point it will have a perfectly random location.
>
> > The question is not the truth value at points well within boundries. It
> > is what truth value to prescribe exactly at the border. I will not try
> > to extend the analogy to your macroscopic example,as feet are not point
> > particles and will overlap the border. Photons, when well behaving, are
> > point particles and do not cause this sort of confusion.
>
> What truth value to prescribe precisely at the border will be given by the
> proper definition; for example, if blue happens when the wavelength is
> greater than or equal to 450, then not blue is anything less than 450. If
> a particle's position is known precisely, it will have a random blueness.
>
> > The gist of my argument is that when attempting to map all
> > statements (in a metalanguage) onto the world, wherever boundries are
> > drawn there will be fundamentally unavoidable ambiguities. At some
> > point (in this case at the points of 540 and 560 nm), it will be
> > impossible to give a truth value of either true or false. I suppose if
> > you insist on maintaining the Tarskian diagram, you could say "le laser
> > est bleu' is Truly True or False, but I don't think this is what Tarsky
> > had in mind! If you find multivalence truly abhorrent, I find no
> > objection to making a Tarskian diagram with three Truth values, True,
> > False, and Ambiguous (or Not Applicable). However, as I pointed out
> > above, the uncertainty principle would make Ambiguous the correct Truth
> > value at every point (each photon may or may not be blue). While
> > accurate, it doesn't do much good to anybody.
>
> Not ambiguous, UNCERTAIN. That's why they don't call it the ambiguity
> principle and why they DO call it the uncertainty principle. And despite
> the fact that the blueness will be completely random, it WILL be either
> blue or not blue, despite the fact that we cannot measure which while we
> know the position perfectly.
Yes, while the frequency of each photon cannot be measured with certainty, you
can measure their average wavelegth with a certain amount of precision, with
variance depending on the accuracy of your instrument. If the accuracy of your
instrument that measures the average wavelegnth is less than +/- 1 nm, then you
have created a bivalent situation, where there is Truth and Falsity.
>
>
> > You could get around the problem of Ambiguity by redefining blue in
> > terms of complex numbers, but your new defintion would not bear any
> > relation to what anybody else thought of as blue.
>
> I presume you are alluding to the Shroedinger equation, by which you would
> say that something was blue if the probability function of its momentum
> fell within certain well-defined boundaries. Interestingly enough, you
> are right about the fact that it would not bear any relation to what we
> call blue, but not because it involved complex numbers.
>
> The easiest way to explain this is with something like the Gaussian
> distribution, better known as the bell curve. Suppose for a moment that
> you had a probability function which looked like the bell curve. Its
> width at the points of inflection would be the standard error (which is
> actually the value which is used in the Heisenberg uncertainty principle).
> Using it, we are accustomed to saying that a particle's position, momentum
> or whatever is given by the position of the absolute maximum value on this
> graph, +/- the standard error. 490 +/- 5 nm, for example.
Yes, chromaticity graphs typically measure the wavelength of light usually by
the peak of the bell curve. Given that you have the vast majority of diffused
light in the sky generated by nitrogen, and some oxygen, as well as particulate
content (I doubt that Tarsky thought much about smog), then you can pretty much
depend on the distribution bell curves for these two gasses overlayed and summed
to determine the actual wavelength seen, depending on the angle of diffraction
the sun is at relative to your present point on the earth. Given the sun's angle
and the diffusion distribution curves, you can determine with quite a bit of
precision whether or not the sky is indeed blue.
>
>
> However, look at the bell curve again. It is not bounded by these points;
> rather > 80% of its area falls within these boundaries; there is a small
> chance that the particle, as described above, should have a wavelength of
> 400nm, an even smaller chance that it should have a wavelength of 600nm,
> and a remote but non-zero probability that it should be 2nm. If we were
> to say that a particle was blue if its probability function met certain
> well-defined criterion, then we would have to say that the above random
> particles would be blue, despite the fact that their wavelength would
> agree much more with purple or even red. We'd still call it blue,
> however, because the probability of it having a wavelength that small or
> that large was small. In other words, an unlikely red is blue.
The human eye tends to make its 'color' determinetion based on the peak of the
bell curve, or in the case of multiple substances, like nitrogen and oxygen
having offset bell curves with differing amplitudes, its a matter of averaging
the two peaks together....
-- TANSTAAFL!!! Michael Lorrey ------------------------------------------------------------ mailto:retroman@together.net Inventor of the Lorrey Drive MikeySoft: Graphic Design/Animation/Publishing/Engineering ------------------------------------------------------------ How many fnords did you see before breakfast today?
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