# Heisenberg Uncertainty Principle and Simultaneous Measurement

Heisenberg says if I measure particle Z’s momentum at t, I can’t know it’s location at t. So why can’t I enlist Jones: he’ll measure Z’s location but remain ignorant of its momentum of course; I’ll measure Z’s momentum and won’t know where it is. Put our results together: voila! Uncertainty fought!

There’s gotta be something wrong with this.

You are talking about extremely small particles, way below microscopic size. The chances that you & Jones would happen to be working on the same particle is vanishingly small, even nonexistent.

Okay, what if we don’t fight the hypothetical.

It’s not about you personally knowing both; it’s about the measurements themselves. Both measurements are still being performed, and so the Uncertainty Principle still applies.

What does “simultaneous” mean? How are you and Jones coordinating your efforts?

How do you arrange for his measurement not to affect yours, and for yours not to affect his? Classically, measurements alter the quantity being measured: that’s true even absent QM. You and Jones are going to be interfering with each other’s work.

Also, you can take the whole affair, you, Jones, and the particle as a single “system,” and that system is still governed by the rules of QM. All systems are!

Chronos is of course right, though I would phrase it differently: the uncertainty principle is saying that the precision with which a particle’s position and momentum can be simultaneously defined is finite. No matter how much I know about a system, it isn’t possible to simultaneously know position and momentum to arbitrary precision, because the system itself cannot have both arbitrarily definite position and arbitrarily definite momentum.

I think Trinopus may be at least in part confusing the uncertainty principle with the observer effect (that the outcome of an experiment alters the properties of the thing being measured) but they’re not the same thing at all. Even if you had a magic experiment that didn’t alter the properties of the thing you’re measuring, you would still fall prey to the uncertainty principle.

The uncertainty principle has nothing to do with a measurement affecting the particle being measured. (The latter is sometimes called the observer effect, but it’s a completely separate principle from the Heisenberg principle.) The Heisenberg principle says that when a particle’s position, for example, is measured with a certain degree of precision, it no longer HAS a well defined momentum, and vice versa. That is, the product of the uncertainty in its position multiplied by the uncertainty in its momentum must always exceed a value related to Planck’s constant. It has nothing to do with the limitations of measurement technology, and it’s not affected by separating the measurements into separate events; it’s a fundamental statement about the wave nature of reality. If you measure the position very accurately, the particle has no momentum for Jones to measure.

(Partially ninja’d by g8rguy)

I would make a pun about the ΔE Δt relation, but I can’t find the energy.

But we want to hear it anyway! … hey, wait a second…

So I was driving along the road with Heisenberg. I looked at the compass display in my rear-view mirror and said “Hey, look, we’re heading due west at 60 mph!” and he yelled “Great, now we’re lost!”

Heisenberg is driving, and he gets pulled over by a cop. The cop says, “do you know how fast you were going?” Heisenberg says “no, but I know exactly where I am.”

One thing to note is that in quantum mechanics, there simply exists no observable corresponding to a simultaneous measurement of position and momentum (a simultaneous precise measurement, that is; there exist generalized ‘unsharp’ measurements corresponding to acquiring simultaneously some information about both). So there just is no experiment where both you and Jones simultaneously measure the particle (and if there were, you could do it alone—in fact, your and Jones’ apparatuses can always be considered a single apparatus).

Some points on the way measurement is formalized in QM. Basically, one tenet is repeatability: if you make a measurement (say, of position) X, then if your make that measurement again, you’ll get the same result. So measuring ‘XX’ really is just the same as measuring ‘X’. Furthermore, there are quantities whose measurements are compatible: i.e. there’s some quantity Y such that measuring that quantity doesn’t influence the outcome of an X measurement—in particular, measuring the sequence ‘XY’ gives the same outcomes as measuring the sequence ‘YX’, and measuring ‘XYX’ will give the same outcome for X twice. In this case, there exists a joint measurement M = XY (= YX) that yields full information about both X and Y.

But certain measurements can’t have simultaneous definite values. In particular, for position X and momentum P, in general, measuring ‘XP’ won’t give the same answer as measuring ‘PX’, and a measurement of ‘XPX’ generally won’t yield the same value of X twice. One often glosses this as ‘the P-measurement influenced the value of X’, but this is really only sort-of right. What’s true is that any given quantum system can only have a sharply defined X or P; but, due to the above requirement of repeatability, it must have a sharply defined X immediately after an X-measurement, and a sharply defined P immediately after a P-measurement.

So if you follow an X-measurement, after which the particle has a sharply defined X-value, with a P-measurement, it must have a sharply defined P-value afterwards, and consequently, can’t have a sharply defined X-value anymore. But that shouldn’t be confused with the P-measurement changing the X-value, as in directly disturbing it: even if we arrange things in such a way that no interaction with the particle is necessary for a particular measurement (which can, for instance, be done using entanglement), we still must describe the system afterwards as having an undefined X-value, even if no physical disturbance has taken place.

Now, for any quantities such as X and P, which aren’t compatible, there also doesn’t exist a joint measurement M as above—since that would have to be equal to both XP and PX, which are, however, generally different. So there just is no joint experiment that Jones and you can perform—either Jones’ learning of P will invalidate your measurement of X, or the other way around.

In fact, your thought experiment is a precursor to the famous Einstein-Podolski-Rosen (EPR)-experiment, which has the added wrinkle that to rule out the possibility of physical disturbance, the position measurement is carried out on one part of an entangled system, which allows to infer the position of the other, while the momentum measurement is carried out on the other part directly, thus apparently yielding both position- and momentum-information. While in this case, there is a joint description of the two measurements (although it would be more accurate to say that if both occur in spacelike separation, there’s no fact of the matter regarding which occurred first), it’s still inaccurate to believe that both quantities could be simultaneously determinate for the system—effectively, the entanglement between both systems means that we can infer the value for one of the quantities from the measurement on the other particle, which then again renders the other quantity indefinite.