From: Daniel Fabulich (daniel.fabulich@yale.edu)
Date: Sat May 30 1998 - 15:50:42 MDT
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. 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.
> 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.
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.
We don't need complex numbers for this, since the probability function is
actually the square of the Schroedinger function (which is the one with
the complex numbers), but it DOES give us some pretty non-intutive
results. For that reason, I'm perfectly fine with saying that red is
bounded by particular wavelengths, rather than particular probabilities,
and that sometimes the redness of a light is truly random.
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