Originally Posted by

**Ken G**
Let's just do why it's hbar, the 1/2 is technical and really depends on how one defines the uncertainty in x and p. Also, note there is no explanation for why hbar has the value it does, that's a matter of pure observation. But we can understand why there is such a thing as hbar, and why it shows up in that relation. It all begins with something called "wave/particle duality." You cannot ask why matter exhibits wave/particle duality, because nobody knows that either, but you can understand that given that it does, we're going to have a HUP.

Many confusing things get said about wave/particle duality, like that it means that matter sometimes acts like a wave and sometimes like a particle, depending on what environment it's in or what questions it is answering. I don't think that's the best way to think about it, because we can't feel like we understand nature by making nature herself sound confused. Instead, just recognize something that was always true that people just never really noticed: *waves are capable of every type of motion that particles exhibit*. So wave/particle duality is not about accepting that waves and particles are two different types of motion and the object sometimes does one and sometimes the other, it is about noticing a type of unification that was always there: all motions we see are motions that waves do. All that differs between "wavelike" and "particlelike" is the size of the wavelength involved-- everything we mean about "wavelike motion" (like doing diffraction and being spread out) corresponds to having a large wavelength, (you get diffraction when the wavelength is not much smaller than the size of the obstacles encountered, and you always have to have at least as much nonlocality as the size of the wavelength), and everything we mean about "particlelike motion" (like following trajectories and well-defined locations) corresponds to having a small wavelength (you don't get much diffraction when the wavelength is much smaller than the obstacles, and if the wavelength is small the wave can be well localized). The "aha" moment is when you realize that *all motion has always been the motion of a wave,* we just never needed to see that until our measurements were precise enough to see down to the wavelength scale of the particles.

So if all motion is wave, then where does the particle part come in? It appears when you ask "what is moving." The answer to that is always "discrete particles, that's what," we just never realized this before because sometimes there are so many tiny particles involved that all we observe is their collective behavior. But we've been down that road many times before-- such as when people were first trying to understand air, and what is moving when there's a wind. Early philosophers sometimes concluded that air, or water or any matter, was a continuous substance that could move around. Others thought the only thing that could move had to be tiny discrete particles. Then came one of the great ironies of physics: it was discovered that the mathematics of waves could explain the behavior of lots and lots of particles, while still holding that individual particles moved differently from waves. Thus was born the fundamental distinction between wave motion and particle motion, and obscured for centuries the fact that wave motion by itself can do particle motion also, the only thing it cannot do is the requirement that everything be composed of discrete elements (discrete numbers of charge, discrete numbers of elemental mass, discrete energy levels, etc.). So the best way to think of wave/particle duality, I argue, is particle handles "what it is", and wave handles "how it moves."

Now we are at the doorstep of the HUP, because if you want to have waves tell particles how to move, you need to connect the momentum of the particle (the fundamental characteristic of a moving particle) to the appropriate wavelength (the fundamental characteristic of a wave). That connection is called the "deBroglie wavelength," where wavelength times momentum equals h, a constant Planck had encountered with light and deBroglie extended to a fundamental relation for all particles. So the wave/particle

duality that had recently been discovered for light (culminating in Einstein's Nobel prize for the photoelectric effect) was extended to all particles, and wave/particle duality was born.

So we see where h gets into the picture, it's in the connection between momentum and associated wavelength for a particle. How does that give us a HUP? It's a simple consequence of how precisely one can associate a wavelength with a wave. The quintessential wavelength is the distance between peaks in a sine wave, but notice that a sine wave extends over an infinite domain, and that's what lets you be very clear what it's wavelength is-- no matter where you look, you see that same distance between peaks. But if the wave had only, say, 3 peaks, and then trailed off in both directions, you only see that distance over part of the wave-- over the rest of it, you see something different, you see a different way that the wave is changing with distance. So you are uncertain what wavelength means when there are only 3 peaks. To see this more quantitatively, perform what is called "Fourier transform" of the wave-- if it's an infinite sine wave, you get a narrow sharp peak at the exact correct wavelength, but if it only has 3 peaks and then trails off, you get a much broader Fourier transform. The ratio of the spread of the Fourier transform (which we can take as the "uncertainty" in the value of the wavelength) to the central value of the wavelength is characterized by one over the number of peaks in the sine wave before it trails off. Another way to say one over the number of peaks is the ratio of the spread of the wave in physical space, to its wavelength, right?

Now we have all the ingredients of the HUP. You simply say that for any wave, its spread in *wavelength* space, divided by its wavelength, is about the same as its wavelength divided by its spread in *physical *space, because that's how Fourier transforms work. Then you tack on the concept of the deBroglie wavelength, to replace the wavelength with h over the momentum. Write out everything I just said as an equation, and it looks like delta lambda / lambda ~ lambda / delta x ~ h / (p * delta x). Since delta lambda / lambda is, by the deBroglie relation, the same thing as delta p / p, this gives us the HUP, to within factors like 4pi that come from the more technical aspects of Fourier transforms and what we are using to quantify the concept of uncertainty. It's all a consequence of wave/particle duality, and the mathematical properties of Fourier transforms-- which physically mean that there is always a connection between how localized a wave is, and the uncertainty in its wavelength.