Heisenberg's uncertainty principle and
the musician's uncertainty principle
Heisenberg's Uncertainty Principle follows from a classical
result, which is at least as old as Fourier. Here, we introduce
it via the Musician's Uncertainty Principle (tuning can be less precise in short notes). This page explains
the classical result and uses sound files to demonstrate it.
It then shows how the Heisenberg's Uncertainty Principle follows
directly from the classical observation, plus the observation of quantisation. Finally, this leads us to some philosophical implications.
When musicians tune up, we listen to the note for a long time
so that we can adjust the frequency precisely. We tune by removing
beats, which are regular pulsations of loudness produced by
notes that are nearly in tune. (See What
are interference beats?) If the frequency difference is
one hertz (one vibration per second), then you hear an interference beat every second (demonstrated
below). If the frequency difference is two hertz, then you hear
two interference beats per second: the difference between the
frequencies of the interfering sounds equals the number of beats
per second.
How long do you need to determine whether two notes are
in tune? Listen to these
sound files, starting with the shortest, and
continue until you can hear beats.
Your results might depend upon your hearing, how much background
noise there is, and how hard you concentrate. But you probably
found that, when the frequencies differed by 3 Hz, you
needed (very roughly) about a third of a second. When they differed
by 1 Hz, you needed more time. So, roughly speaking, if
the frequencies differ by Δf, then
you need a time of 1/Δf to notice.
In other words:
Δf.Δt
> ~ 1 or, in non-mathematical
language:
(time taken to measure f) times (error in f) is on the order
of one, or larger. This result follows simply from the work of Joseph
Fourier, more than 200 years ago. In this context, let's call it:
The musician's uncertainty principle. Because musicians
know this, qualitatively at least. If the chord is short,
or if you are playing a percussive instrument, the tuning
is less critical. In a long sustained chord, you have to get
the tuning accurate. And of course oboists in orchestras play
notes for up to tens of seconds while the other instruments tune
carefully before a concert.
So far this calculation is just an order-of magnitude. One
can imagine doing a litte better than Δf.Δt > 1 by careful measurement. (Have
a look at the diagrams on What
are interference beats?). The uncertainty principle is
usually written with an extra factor of 2π:
it takes about one radian rather than a whole cycle. So:
(time taken to measure f) times (error in f) is greater than
about 1/2π.
Heisenberg's uncertainty principle
Now in quantum mechanics and atomic physics, the energy of a
photon is hf, where h is Planck's constant. So a measurement
of the energy corresponds to a measurement of the frequency,
and that, as we have seen, takes time. Mutliplying our previous
inequality by h on both sides gives us Heisenberg's uncertainty
principle for energy
(uncertainty in energy) times (uncertainty in time) is greater
than about h/2π, or
ΔE.Δt
> ~ h/2π.
Heisenberg's uncertainty principle for momentum is
analogous. Let's consider the spatial frequency F (which
is defined as the number of cycles per unit distance) rather
than temporal frequency (number of cycles per unit time).
F is just the reciprocal of the wavelength, λ.
The same argument about beats in this case gives (for motion
in the x direction)
The momentum of a photon (or anything else) is p = h/λ
so, multiplying the above equation by h gives
which is usually written as
Werner
Heisenberg won the Nobel prize in 1932.
Practical consequences of the uncertainty principle.
h is very small (6.63×10-34 Js), so the consequences
of the uncertainty principle are usually only important for
photons, fundamental particles and phonons. (See, for example,
this example
using a cricket ball.) There are, however, many physical
processes whose evolution with time depends sensitively on
the initial conditions. (Sensitivity to intial conditions
is sometimes called chaos.) The uncertainty principle prohibits
exact knowledge of initial conditions, and therefore repeated
performances of such processes will diverge. Physicists will
also tell you that one cannot have exact knowledge anyway,
for a variety of practical reasons, including the fact that
you don't have enough memory to record the infinite number
of significant figures required to record an exact measurement.
There is also a discussion of how chemistry depends upon
the uncertainty principle at Why
there would be no chemistry without relativity, which
is part of our site on relativity.
Philosophical consequences of the uncertainty principle.
Some philosophers regard the consequences of the uncertainty
principle as having a more fundamental importance. Their argument
goes like this: if one could know exactly the position, velocity
and other details, one could, in principle, compute the complete
future of the universe. Since one cannot know the position
and momentum of even one particle with complete precision,
this calculation is impossible, even in principle. Some attempt to draw profound conclusions from this simple observation. Most scientists
find this a trivial argument. A memory capable of storing
all this information would be as complex as the universe,
and then the contents of that memory would have to be included
in the calculation, and that would make the amount of information
greater, and that information would have to be stored.....
We rather point out that all of that information is actually
contained in the universe which, as an analogue computer,
is computing its own future already.
A list of educational
pages by Joe Wolfe |
