Heisenberg's uncertainty principle and

the musician's uncertainty principle

How long do you need to determine whether two notes are in tune? Let's find out experimentally. Download these sound files and listen, starting with the shortest file and continuing until you can hear beats.

400 & 403 Hz 0.08 s

0.17 s

0.33 s

0.67 s

1.00 s

2.00 s

5.00 s 400 & 401 Hz Sound files courtesy John Tann 0.08 s

0.17 s

0.33 s

0.67 s

1.00 s

2.00 s

5.00 s



Δf.Δt > ~ 1 or, in non-mathematical language:

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 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

(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)

ΔF.Δx = Δ(1/λ).Δx > ~ 1

Δp.Δx > ~ 1

Δp x .Δx > ~ h/2π. (uncertainty in momentum) times (uncertainty in position) is greater than h/2π.

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 fashionably 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