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Let's Use Acoustic As An Example Of Wave Packet.

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Posted by Alexander on September 21, 2001 20:01:57 UTC

Let me explain the same thing (uncertanty principle for waves) without ising Fourie images (although Fourie transforms show uncertanty principle in one string.)

Suppose you have a sound corresponding to C note (f=256 Hz). Speed of sound in the air at room temperature is about 340 m/sec, so wavelength of this note then is about w=c/f = 1.33 m. But let's take a close look at how we define and how we measure what we call "wavelength".

If you let a tuning fork to sound for 1 second, you then produced a wave train (packet, bunch, etc.) with 256 waves in it. Wavelength is defined as the total length of this train devided by the number of waves it has. Overal length of this train is 340 m, so wavelength is 340/256 = 1.33 m, and this train moves with speed 340 m/sec away from the tuning fork.

{By the way, if it moves in 1-D space (say, long pipe it travels in, or a long string) the amplitude of the wave stays the same as it travels - it does not lose energy with distance because there is no no divergence (spread) of this energy. In 2-D space (rubber sheet) energy spreads around in all directions thus energy dencity drops as 1/distance, in 3-D space - as 1/(distance^2), in 4-D as 1/(distance^3) and so on. This is why in 3-D space fundamental interactions (gravitation and e/magnetic) obey inverse square law which is just a geometry of 3-dimensional space.}

Now, lets decrease the time we allow a tuning fork to oscillate from 1 sec to 0.998 sec (so, we reduced a train length from 340 m to 339.3 m, or from 256 waves to 255.5). How do we define and how do we measure the wavelength now? Well, because we are not sure if we still have 256 waves in our train, or it is already 255 waves, our definition of a wavelength will be anything from L/256 to L/255, so it is about 1/256 UNCERTAIN. So, having a train with N wavelength is equivalent to defining its frequency with 1/N uncertanty. Infinite train therefore has exact wavelength, but the shorter "chunk" you cut out of it, the less certain will be the wavelength of oscillations in that train.

All this equally apply to frequency - it is also 1/N uncertain (thus,it is exact only for infinitely lasting oscillations, and spread around some central frequency for a short lasting note).

Momentum ("amount of motion in given direction) of a wave is given by so called wave number k=2pi/w. So, if the train has length equal to N wavelength (L=Nw), the uncertainty in its momentum is also 1/N (N=100 oscillations result in 1% uncertain momentum, N=5 in 20% uncertain and so on). So, delta(k)/k = 1/N. We can write it also as delta(k) = k/N

On the other hand, how do we define a "position" of a wave train? "Middle" of a train? But what if a train is more "loud" (waves are more intense) in the very beginning (or at the very end)? All we can do to define the position of a wave train is to say that it is somewhere within its length L. Therefore, uncertanty in its position by definition is about L.

Thus, the uncertanty principle for any wave train: delta(x)delta(k) = Lk/N = wNk/N=wk=2pi=6.28

In quantum mechanics a quantum wave with the wave number k has the momentum p=khbar, which can be written as k=p/hbar. Therefore the wave train uncertainty principle becomes delta(x)delta(k) = delta(x)delta(p/hbar)=2pi, or delta(x)delta(p) = 2pihbar = h.

Some may get impression that the UP for waves is just position and momentum "definition" flaw, some - that we just have to come up with better way of measuring wavelength and then uncertainty principle will vanish. There are tons of near-technical literature about our imperfect mathods and tools and suggestions how to improve them. There are tons of philosophical literature about "meaning" of this uncertainty and about consequences of UP for sociely, religion, etc.
But it is nothing more than just mathematical property of any wave train - its momentum is tied to its length (being the Fourie-image of length, and its energy is tied to its dime duration (also Fourie images of each other). It is like the relationship between the volume of a gas with given number of atoms in it and the density of gas - the larger the volume, the less the density.

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