Beating the Uncertainty Principle

Most recent answer: 09/12/2013

Q:
This morning I asked myself a simple question and was annoyed that I couldn't come up with the answer - every time I thought I had one I could turn things around and talk myself out of it. The question is pretty simple, though I'm not sure I can state it as elegantly as I'd like. I don't *think* it's hopelessly naïve, but one can never be sure... Anyway, the question is this: The Heisenberg uncertainty principle limits what you can know about complementary variables; I believe it applies to "now"; does it also apply to the past? That is, is there a way I might (in principle) make a series of observations now that would let me determine what the position and momentum of a particle were (say) hours ago, to a greater precision than the uncertainty principle should allow? I think the answer is no, but I can construct counterexamples that challenge this. I think I understood the issue (years ago) by thinking of the particle as a wave packet - that is, that the uncertainty principle said more about reality (the particle not having well-defined position, momentum) than about any measurement issues. The counterexamples I run in my mind all involve some form of later 'wave function collapse' that seem to provide enough data to effectively overconstrain the problem. At the risk of going on too long, here's one sort of thought experiment - a neutron hits a large nucleus (elastically; the initial nucleus position/momentum known as optimally for this problem as possible), which splits in two equal halves; I later observe the two halves. Perhaps I measure the position of one and the momentum of the other; it seems intuitively that, along with what I knew about the nucleus at the outset, I might be able to reconstruct enough about the collision that I could compute the position and momentum of the incoming neutron to a precision greater than the uncertainty principle would allow. I would not have been able to do this at the time of the collision, but looking back into the past it seems I have more and more evidence with which to work. On the other hand, if I indeed could work out (say) what the position and momentum were in the past, those would look a lot like the "hidden variables" of the sort Bell's Theorem refutes. But not necessarily; the fact that I could come up with those values now might just be a sort of labeling, without deep meaning; after all, there is no way I can go back and test it. If I had to generalize the problem, I could imagine interaction A leading to interactions B, C, D, etc. each of which lead to further separate interactions, in a sort of logical tree. If there is anything like wavefunction collapse at any of the leaves of the tree, it would provide information about that branch going back to A, and thereby partial information about the entire tree. But this can't happen dynamically, as those events are in the past. (I reference 'wave function collapse' specifically meaning that I get a single measured result, such as position or momentum; I'm happy to admit I've never been happy with that concept, or what in detail might be happening). It is certainly possible that if I actually calculate the details of the above example I might find I really can't come up with past values for a particle that violate the uncertainty principle; on the other hand, as reality moves forwards, the impact of past states spreads, providing more information to work with. I can't help thinking I'm overlooking some critical points. After a bit of Googling to see if there was a definitive answer out there, I thought I'd find someone to ask - I would think that this is the sort of thing that would have been settled long ago, if it were ever an issue.
- Bryan Bentz (age 53)
Stonington, CT USA
A:

Hi Bryan,

That's a good question. I think the simplest case to understand is basically the one you pointed out. Say a source particle decays into two photons; one shoots off generally upwards, while the other shoots off opposite. Of course, if you measure one photon's position on a film screen a meter away from the source, you will get a single blip corresponding to a rather definite location. Since you know almost exactly where the photon started (at the source, whose location can be measured before the emission), and you know almost exactly where the photon ended up (at the blip on the film), it would seem obvious that the particle's momentum was directed along the vector connecting these two points. As such, it seems that you precisely know what the particle's almost exact position and momentum were before it hit the screen.

That would definitely violate the uncertainty principle. Furthermore, if you repeat the experiment many times, you will find the photon in a new location on the screen each time, corresponding to a new momentum vector. You might think that there is some unknown difference between each trial, which tells the photons to have different momenta upon each emission. Such an instruction, or "hidden variable," is in violation of Bell's theorem, as you know, and has been experimentally disproven.

Instead, standard quantum theory says that the spread in your position and momentum measurement results IS an accurate portrayal of the photon, which was emitted into a superposition of many momenta, with mean values "up" and "down" for photons 1 and 2, respectively. The position of the particle also has an uncertainty, since the beam has a spread in space. When you do the position measurement, you randomly collapse the position of the particle, and you don't learn very much about the momentum of the particle. *

In no case, even if you use past information or measurements or entangled particles, can you beat the uncertainty principle. This is because, as you suspected, the principle doesn't apply just to human knowledge, but is actually a property of the physical state itself.

Hope that helps,

David Schmid

​* As I understand it, the detection of the photon in a little region gives you very much the same sort of information about its past state as knowing that it came from a little region told you about its future state. The source info told you that the future state would be spread out leaving the source, in a simple case spherically spread out. The detection tells you that the past state was spread out, e.g. spherically, converging toward the little detection region. So now you have both types of info about the state that was present between emission and detection. The combination describes a state that spreads out from the source and then converges toward the detection. 

But wait, you say. That's not a possible trajectory of any classical particle. It's not even a combination of many different classical trajectories. That's right, because there is no such thing as a classical particle. As David explained above, violation of the Bell Inequalities tells us that that quantum state is all that there was. Mike W.


(published on 09/12/2013)