From: Dejan Vucinic (dejan@mit.edu)
Date: Wed Apr 16 1997 - 13:40:51 MDT
John K Clark:
> Mass, energy, momentum, electric charge, spin, charm, and color,
> are all digital quantities.
Me:
>> Digital they aren't, discrete is what you probably meant. And
>> discrete they are not either. Spin and charge of particles are
>> observed as discrete values, the discreteness of color charge is
>> deduced indirectly, but mass, energy and momentum are observed
>> as continuous.
> Max Planck started the entire revolution by proposing that energy is
> NOT continuous, everything followed from there. Take a beam of light
> for example, it has momentum nhf, h is the Planck constant, f is the
> frequency of the light, and n is an INTEGER, the number of photons.
> Digital.
The formula is correct, but your conclusion is not. While n is an
integer, f is not. That means that a "beam" of light consists of
a discrete number of quanta (as opposed to a continual wave), but
the individual energies of those quanta can have any value from
a continuous spectrum. In a special case of coherent light, we
engineer the source in such a way that all the quanta have the same
energy, for example by making sure all of them are emitted from a
certain transition in a certain kind of atom. However, if you move
the coherent source relative to the observer, you can change the
observed frequency (energy) of the quanta by a non-discrete amount.
This is called the Doppler effect. It's even simpler to make the
frequencies of quanta in the same beam different: just heat up the
source!
> If you want to find the momentum of an individual photon, hf, you can
> find it by using the formula hN/X , h is the Planck constant again,
> and N is the number of wave crests the photon makes over a distance X.
> Digital.
Another wrong conclusion. One: N is not an integer. A photon that
has one "crest" over one meter has one point three crests over one
point three meters. Two: the number of crests depends on the frame
of reference. If you measure five crests over one meter, and I am
moving relative to your rest frame, I may measure five times square
root of two crests over one meter. But now I see how you got to
that conclusion:
> Of course, there is a basic randomness built into the fabric of
> the universe, so if we were to actually perform this experiment
> when we got near the end of distance X it would not be entirely
> clear if we should include the last wave crest or not. We could
> minimize this problem and get a more accurate measurement of
> momentum by making X bigger, but then you'd know less about
> the position of the photon, X. Momentum and position have an inverse
> relationship, the more you know about one the less you know about the
> other, however randomness is NOT a continuous quantity, in fact it
> is as far from that concept as it is possible to get.
You're now drifting toward the question of interpretation that has
been a hobby of every self-respecting quantum mechanic since the
thirties. You're trying to dump more information into the problem
in order to try to make it look more "intuitive". Counting crests
of photons is a mental crutch. In reality, what you can do is
measure the energy of your pet photon. Unfortunately, in the
process you're destroying the photon itself, so the accuracy of
your measurement is not constrained by the Heisenberg inequality,
but rather by the construction of your apparatus. You can even
measure with some accuracy where in space did the photon convert,
but that doesn't tell you whether the photon was a little ahead or
a little behind that point in space at the moment of destruction.
And you are right in that randomness is not a continuous quantity.
In fact, it's not a quantity at all, it's a concept people have
trouble imagining because it's not "intuitive". However, an
incredible mountain of evidence suggests that quantum mechanics is
how universe works, so the right attitude imho is to convince
oneself that there simply are questions that have no meaning within
the confines of the working theory, and that their meaning emerges
at some other level. Isn't complexity fun? :)
> If you want to find the energy of a photon multiply the digital
> momentum by a constant, the speed of light. Digital.
No, it just says that our historical choice of units for "energy"
and "momentum" was arbitrary, so we invented a "constant" to make
it sound less embarrasing. That's not a very useful observation
if you build bridges for a living, but in the Universe and in my
current profession c == h == 1.
Regards,
--dv
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