As we started the last week of the advent calendar, I was trying to map out the final days, and was coming up one equation short. I was running through various possibilities– the Dirac equation, Feynman’s path integrals, the Standard Model Lagrangian, when I realized that the answer was staring me right in the face:
This is, of course, the Heisenberg Uncertainty Principle, saying that the product of the uncertainties in position and momentum has to be greater than some minimum value. Strictly speaking, this should’ve come before the Schrödinger equation, if we were holding to chronological order, because Heisenberg got there first, but it’s also the equation that gave this blog its name, so I held it for last.
So, what does this mean?
What does this one mean? A lot of popular descriptions of the principle put it in terms of measurement perturbing a system, saying that any attempt to do a better measurement of the position of a particle will necessarily introduce uncertainty in the position, and so on. You can even get the basic form of the relationship right, by thinking in these terms, and using a little knowledge of optics.
This is a little problematic, though, in that it presumes that the particle has a well-defined position and velocity, even if those things aren’t known to the experimenter. While there are ways to look at quantum theory where that’s a sensible statement, the underlying mathematics point to a different way of thinking about it, which is to say that these quantities are simply not well defined to begin with. You can’t do a perfect measurement of the position of a particle not because of practical limitations on your measuring apparatus, but because it doesn’t make sense to talk about the position of a quantum particle having a definite value at all.
This, of course, brings with it a whole host of problems, and the probabilistic interpretation of the theory turned off a lot of physicists, including people like Einstein and Schrödinger who had helped launch the theory. But if you’re willing to roll with it, it’s an incredibly cool way of looking at the world. Which is why I appropriated it for the name of the blog. It’s also useful as a reminder that you can never really know everything about anything, which is important to keep in mind.
This brings us to the end of our physics advent calendar. Newton’s birthday is tomorrow, but we’ll all be too busy to blog, let alone solve equations. But whether you mark the date with a Christmas tree, a menorah, or an obsessive personal quest for alchemical secrets, take a moment on the 25th to appreciate physics, and all the cool things it does for us. And try to carry that appreciation through the rest of the year– for that’s the real meaning of Newton’s birthday…
OK, tx Chad. I’d like to add to why we can’t violate the HUP, and some other thoughts. The best reason in the opinion of many, has to do with wave mechanics and Fourier analysis. A particle can be represented as a “wave packet” that is a superposition of various pure sine “momentum waves” per p = hν (that’s “nu” for frequency.) The WP represents chance of finding the particle (indeed! – and furthermore the whole wave represents the entire mass-energy, so IMHO can’t localize all of it at multiple locations, regardless of the excuse given …)
So, if the WP was a pure momentum, the equivalent wave would be endless: no specific position. If you can find a specific location, then math – the Fourier analysis – demands a certain spread of frequencies and hence momentum. Hence the HUP is not ad hoc, it derives by necessity (maybe not a simply as I describe) from self-consistency and math, once the basics of wave mechanics are established.
And yes indeed, that necessity is not based on needing to disturb the particle. Yes, I know about “Heisenberg’s microscope” and the dependence of resolution on frequency of photon. However, suppose for example we viewed little bits of optically active material with polarized light: the bits would turn the angle of polarization, but not be pushed out of the way (not even net change in angular momentum.) We could see them under crossed polarizers. (Also, consider oil-immersion’s increased resolution at same frequency of light.)
Finally, note the curious reference of the frequency-momentum relation to a give single particle. That may not seem problematical at first, but remember Galileo’s argument against Aristotle’s view of falling bodies (speed proportional to mass, or was that a caricature.) What if you tied two bodies together with a string, then are they like “one” – not matter how thin the string – or still “two”? It didn’t make sense for a physical action to depend on our Sorites-like categories of “single object” versus “composite” etc.
Yet the FMR is just like that! So we can ask, how “connected” do particles have to be, to count as “one” for defining the matter-wave frequency going with the appropriate collection of mass? I presume and from discussions that interference and harmonics effects take care of the logical question, anyone else have insight or scoop?
Merry Christmas Eve, etc., everyone; and Happy Newton’s Birthday (I had forgotten that coincidence, REM however it’s per “old system” calendar – yet just as easy to appreciate the great man’s spirit.)
Too bloody right the HUP has got nothing whatsoever to do with ‘disturbances’ or ‘simultaneous’ measurements (whatever they are). 😉 http://www.phys.tue.nl/ktn/Wim/qm12.htm#Martens
Sorry, left out “c” in my relation above. It should read, p = hν/c. Same point. In any case, the cool way is to write E = Ä§Ï and p = ħk.
This has nothing to do with Planck’s constant. But here are two variable which are also related in a similar way, at least for me, and probably for other people.
Speed of reading versus comprehension and retention.
Fast reading is OK to figure out what is it about, superficially. But it is not useful for learning something new and difficult.
Ludwik Kowalski (see Wikipedia), is the author of this autobiography:
http://csam.montclair.edu/~kowalski/life/intro.html
Neil Bates wrote (December 24, 2011 10:58 AM):
> So, if the WP was a pure momentum […]
This may turn the page on the tired calendar motto
“ […] it doesn’t make sense to talk about the momentum of a quantum particle having a definite value at all “.
> […] remember Galileo’s argument against Aristotle’s view of falling bodies
> […] how “connected” do particles have to be, to count as “one” for […] the appropriate collection of mass?
First note for a state Ï of definite momentum p (and Energy E):
d/dx[ Ï / Exp[ i (p · x – E t) / â ] ] = 0 and
(â / i) d/dx[ Ï ] = p Ï.
Equating “Exp[ i (p · x – E t) / â ] ]” and “Exp[ i (k · x – ν t) ] ]” therefore
p â¡ â k, or
p ⡠h / λ.
From the “wavelength λ” (evaluated for instance by measuring an “interference pattern”) follows the value “p” as characteristic of one “momentum quantum” in the experiment under consideration.
This may have some similarity to “Aristotle’s view“:
“The shorter the wavelength, the larger the momentum quantum”.
However, numerous quanta of equal momentum would have to be considered together to obtain a useful “interference pattern”.
That seems closer to “Galileo’s argument“:
“Doubling the given (already sufficiently large) number of equal quanta, with an otherwise unchanged setup, still yields (at least roughly) the same value λ from the resulting pattern.”
And this also applies if individual entries in the “pattern” turn out to be (due to) systems themselves; such as (due to) “composite particles”.
I think you guys need to step back a bit. You say you cannot violate Heisenberg, but what you do not say is that quantum mechanics is a theory of MEASUREMENT. Hence the uncertainty relations put a nice big limit on our ability to measure–nothing else really. But Nature does not give a hoot about us, our measurement and our conclusions. Our experiments are just another interaction and the problems are ours, not Nature’s. Also those stuck on QM being a complete theory and therefore accept a probabilistic interpretation not only forget QM is about measurement only, but also fail to realize that all probabilities are a statistical build up from single events. The fact that those single events have not yet been properly identified is not Nature’s fault but ours.
I think you guys need to step back a bit. You say you cannot violate Heisenberg, but what you do not say is that quantum mechanics is a theory of MEASUREMENT. Hence the uncertainty relations put a nice big limit on our ability to measure–nothing else really. But Nature does not give a hoot about us, our measurement and our conclusions. Our experiments are just another interaction and the problems are ours, not Nature’s. Also those stuck on QM being a complete theory and therefore accept a probabilistic interpretation not only forget QM is about measurement only, but also fail to realize that all probabilities are a statistical build up from single events. The fact that those single events have not yet been properly identified is not Nature’s fault but ours.
Bryan Sanctuary wrote (January 1, 2012):
> quantum mechanics is a theory of MEASUREMENT
Of course. (This usually goes without saying, but bears repeating.)
> But Nature does not give a hoot about us, our measurement and our conclusions.
Apparently that’s how some put it.
But if you succeed in telling your friends (properly) what you’ve done then they, at least, may give a hoot about what you’ve found …