# Uncertainty principle and multiple observers

My understanding is that an observer can measure the precise location of a particle so long as the corresponding uncertainty in momentum measurement is not an issue and vice-versa.

Say there is such an observer, interested in the precise position of a particular particle. Now, consider a second, independent observer, unbeknownst to the first, who is trying to measure the exact momentum of the same particle without caring about the position. As a thought experiment, we assume that the two observers are somehow able to access the same particle at the same time in some way without being aware of each other.

Can both observers get their desired results?

• This might be relevant: tandfonline.com/doi/abs/10.1080/09500349314551761#preview as well as this: onlinelibrary.wiley.com/doi/10.1002/j.1538-7305.1965.tb01684.x/… – Alfred Centauri Aug 27 '13 at 22:02
• @AlfredCentauri I can't get the second article, even through my university, but for the OP's benefit here is a relevant quote from the beginning of the first article: "Heisenberg's uncertainty relation predicts that a quantum-mechanical particle cannot have a precise position and a precise momentum at the same time. According to Stenholm, 'it is, however, possible to obtain information about both position and momentum, if one accepts a fuzzy picture of both within the bounds set by the uncertainty principle'." ... – Michael Brown Aug 27 '13 at 23:03
• (contd.) "A measurement in this spirit may be called a 'simultaneous measurement of canonically conjugate variables', whereby, of course, not the 'true' position and the `true' momentum of a quantum-mechanical system are measured, but both variables influenced by additional noise." (emphasis mine) – Michael Brown Aug 27 '13 at 23:04
• The Heisenberg uncertainty is not about (simultaneous) measurements. See for example my (and other people's) answer here physics.stackexchange.com/questions/102834/… and physics.stackexchange.com/questions/169730/… and similar questions of this type. – Martin Feb 20 '16 at 0:39

They could not - don't think of the observers as people, think of them as experiments. If two individual experiments were taking place on the same particle simultaneously, there's no reason why they couldn't be combined into a singular experiment, and you'd then have an experiment that is able to determine both a precise position and momentum, which is impossible.

• Do we know what mechanism would cause the experiment to fail? For example, if the two measuring devices were tested separately and verified to work to a certain precision, what would go wrong and cause the precision to be reduced for each device when they try to measure simultaneously? Is there some inherent property of the particle itself that could resist precise measurement? – mbeckish Jun 12 '14 at 20:39
• There's no possible way to take a measurement of something without also interacting with it somehow. The interactions from the two experiments would interfere with each other, causing both experiments to lose accuracy. – MattS Jul 5 '14 at 21:30

They could not! The explanation has to do with the nature of what measuring momentum and position involves. First, understand what the uncertainty principle is truly about:
$$\Delta x\Delta p \ge \frac{h}{4\pi}$$
The equation specifically pertains to the uncertainty of each measurement. That is, the greater you know of the accuracy of one measure, the less accurate the other measure will be. That is why trying to know both leads to a fuzzy picture of accuracy. To understand why you can not know both regardless of how many observations you make, consider that the uncertainty principle does not relate to the observers, but the measurements themselves.

If a bee flies around at a high speed, you can "freeze" it to know the position at a specific point in time. However, to know the momentum, you need to track it across some measure of time as the bee moves. Obviously if you freeze it in place to note its position, it doesn't travel any distance for the momentum to be measured accurately. To that extent, the more time you allow it to move, the greater your accuracy of the momentum. So, the more you slow it down, the more you can predict its position, but the less you know about how fast it is going. The opposite is true as well; the more you let it move, the more you know about how fast it is going, but less about where it is at any point in time.

I hope that made sense! The key to understand is that the uncertainty principle is not based on the number or quality of observers, but on the specific nature of what is being measured!

Let me preface with this: I am not a physicist, only an interested party who has thought about the same phenomenon.

I'd like to start by saying that it is possible. Now the two above, longer replies I have read, and, to myself, understand. The reason being begins with suppsoing that the two proposed observers could be left totally independent (say we were using machines of incredible accuracy that could not communicate and thus become the same observer). Now, as above, set one machine to measure position and the other to measure momentum. As mentioned above, the uncertainty principle dictates that the more accurately one measurement is made, the accuracy of the other beings to decrease (uncertainty increases).

Given that each machine is only logging one measurement to the greatest possible accuracy it could achieve, and both do not intend to measure the other property (the machine measuring momentum doesn't measure position and the machine measuring position doesn't measure momentum), then, after they are done taking measurements, you put their logs side by side (assuming that these machines were recording their respective information in sync) you have both the momentum and position of that particular particle over a set amount of time.

I know that the above can be debated with the above statement that the uncertainty principle is not based on the number of observers but the nature of what is being measured. But that is also exactly why it should work. Each observer sees, essentially a different particle, one sees a particle with no momentum and the other sees a particle with no position (only said for the sake of simplicity, believing the accuracy of the measurement to be absolute and thus the uncertainty of the complimentary measure to be absolute), despite, to a third party, to be observing the same particle. Now here's another counter argument: These observers are observing different particles and thus the measurements cannot be compared and treated as if they came from the same particle. However, I would like to mention that it is not these parties (the two observers), who are responsible for taking the measurements, who are comparing the data sets, but a third party who was observing the observers observe the particle, and understands that whatever data comes from those observers is in fact data related to and only to that particle.

This is the part where I pick apart why I said it was 'possible'. And it is for the following reasons: 1) There must be only one possible, observable particle for both observers to measure. 2) You must be able to observe and measure the particle without disturbing the particle's motion. 3) The observers must begin making their measurements at the same time, at the same rate and with the same accuracy.

And because these criteria are hard to meet, particularly number three, it is merely 'possible' (in a theoretical sense) but not practical.

If none of that made sense, please point out any errors or flaws. If your just confused, then ask for clarification.