Anders Sandberg wrote:
>
> Rob Harris Cen-IT <Rob.Harris@bournemouth.gov.uk> writes:
>
> > I see, in the most part. I just don't get the most juicy bit. How was
> > Planck's constant arrived at and declared the limit of all detection and
> > computation, regardless of the system in question? Why is tracking the
> > momentum AND the position so difficult?
>
> Usually you hear heuristic explanations of the kind "you cannot see a
> proton without bouncing a photon from it, and that changes its
> momentum", but the real explanation is based on the math of quantum
> mechanics. I think it has to do with that certain operators do not
> commute with each other (AB isn't BA). Hopefully somebody more versed
> in QM can give the correct explanation.
>
> This is a rather firm limit - somehow I don't think even SIs can edit
> the commutability of all operators...
>
I'll take a shot. The Uncertainty Principle is built right into the theory, and it's an absolute limit, if QM is correct. Just as the speed of light is an absolute limit, if relativity is correct (an important caveat).
According to basic QM, observables such as position and momentum are transformed into "operators". Particles are transformed into "wave functions". In order to predict the outcome of an experiment in which, say, position is to be measured, you apply the X operator to the abstract wave function w, and you come up with a function w(x), on the x (position) variable, that gives the probability of finding the particle at that point x. Similarly, the momentum operator P, when applied to w, would give a function w(p) which gives the probability of finding the particle to have a given momentum p.
Now here's the tricky bit: when you measure an observable, you necessarily change the wave function. Say I have a particle with a wave function that is a nice bell shaped curve centered at the origin. QM says if I measure the position x, there is a range of values that it can take, and there is no way for me to predict the outcome of the experiment beforehand. Let's say I measure the position and come up with a value of 2. *Now*, if I measure the position again right away, QM says that I am guaranteed to get the same value, 2. The only way for that to have been possible is if my original measurement changed the wave function, so that instead of a nice bell shaped curve around the origin, it's now a sharp spike centered at the point x = 2.
The reason I'm guaranteed to get the same result if I measure position again right away is that the X operator commutes with itself. That is, it doesn't matter which I do first: measure the position, or measure the position (obviously, because they're both the same operation!) But the X and P operators do not commute. That is, XP is not the same as PX. The order of the measurements matters. This is defined by the "commutator", an expression which defines this relation:
XP - PX = i(hbar)
Compare this with the commutator of X with itself:
XX - XX = 0
What this means is that if I measure the position and get a well defined value, then my uncertainty about the momentum increases. A nice analogy is the fourier transform of a sound. For a sound, you cannot simultaneously know exactly when the sound occurs, and exactly what it's pitch is. That's because the more well-defined you make the pitch, the more spread out in time the sound must be. Likewise, the more you try to pin down exactly when the sound occurs, the less information you have to discern its pitch. Taken to the extreme, if I nail down the time that a sound occurs to be exactly t = 0, then I have "collapsed" the sound into an impulse wave. What is the "pitch" of an impulse wave? Well, an impulse wave contains all frequencies, in equal measure, from zero to infinity. Thus the pitch would be completely undefined.
In textbooks they often show how the uncertaintly relations manifest themselves by describing attempts to measure the position and momentum in quick succession. And it turns out that no matter how you arrange the experiment, measuring one automatically affects your ability to measure the other. Thus, for example, to measure the position of an electron, you have to bounce a photon off of it. But to measure the position accurately, you need to use a photon with a very small wavelength. But the smaller you make the wavelength, the higher is the photon's energy, and when it hits the electron, it's going to affect it's momentum in an unpredictable way. But these are just manifestations of the underlying theory, which says, unequivocally, that you cannot simultaneously know both observables. The theoretical limit of this uncertainty is defined by Plank's constant, hbar.
The ontology of this phenomenon is still hotly debated.
-- Chris Maloney http://www.chrismaloney.com "Knowledge is good" -- Emil Faber