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Thought experiment: what happens if we measure momentum of a particle so precisely, that the uncertainty of its position becomes absurd?

For example, what if the uncertainty of the position exceeds 1 light year? We know for a fact that the particle wasn't a light year away from the measuring device, or else how could the momentum have been measured?

What if the uncertainty extended beyond the bounds of the universe?

Isn't there some point at which we know for certain the particle was closer than what the uncertainty allows for?

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how can we talk about particles for which we are not certain they are somewhere in the universe ? –  Andre Holzner Jun 13 at 10:32

3 Answers 3

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You assume that you can instantly measure the momentum to arbitrary precision, and this isn't the case.

Let's consider a plane light wave to keep things simple, and suppose you want to measure the momentum so precisely that the position uncertainty becomes exceedingly large. How precisely do we have to measure the momentum? Well the uncertainty principle tells us (discarding numerical factors since this is all very approximate):

$$ \Delta p \approx \frac{h}{\Delta x} $$

For a photon the momentum is $p = hf/c$, so this means we have to measure the frequency to a precision of:

$$ \frac{h}{c}\Delta f \approx \frac{h}{\Delta x} $$

or:

$$ \Delta f \approx \frac{c}{\Delta x} $$

Suppose we want our $\Delta x$ to be one light year, our expression becomes:

$$ \Delta f \approx \frac{1}{1 \space \text{year}} $$

But to measure the frequency of a wave accurate to some precision $\Delta f$ takes a time of around $1/\Delta f$. This is because the frequency you measure is the wave frequency convolved with the Fourier transform of an envelope function, and in this case the width of the envelope function is the time you take to do the measurement.

So the time $T$ we take to measure our momentum to the required accuracy is:

$$ T \approx \frac{1}{\Delta f} \approx 1 \space \text{year} $$

The conclusion is that to measure the momentum precisely enough to make the position uncertainty 1 light year will take ... 1 year!

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Thanks! That explains my "locality" confusion - the amount of time necessary to make the measurement works out to be exactly the amount of time necessary for information to travel from the furthest reaches of the Δx. What about when spacetime itself is smaller than Δx (for example, accurately measuring momentum of a photon in the early moments after the Big Bang). Is it like @doetoe suggests, that the boundary conditions would place a limit on how accurately the momentum can be measured, so that Δx does not exceed the size of the universe? –  mbeckish Jun 13 at 13:35
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@mbeckish: assuming the universe is currently infinite it was laways infinite, even at the moment of the Big Bang. The size of the observable universe started at zero, but expanded at the speed of light thereafter. So I don't think there was ever any way to measure momentum precisely enough for $\Delta x$ to exceed the size of the observable universe. –  John Rennie Jun 13 at 15:43

Consider a measurement apparatus (M.A.) of characteristic size $d$, the uncertainty about the momentum of the measurement apparatus is then $\Delta p_{M.A.} \approx \dfrac{\hbar}{d}$

Now, if the measurement apparatus is measuring the momentum of a particle, the uncertainty about the measured particle momentum is necessarily greater than the uncertainty of the measurement apparatus momentum:

$\Delta p \geq \Delta p_{M.A.} $

So, finally, you must have $\Delta p \geq \dfrac{\hbar}{d}$

That is to say, you cannot measure a particle momentum with a precision greater than $\dfrac{\hbar}{d}$ with a measurement apparatus of characteristic size $d$.

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Do you have a citation handy for the first equation? A quick google search didn't yield any information on the concept of "characteristic size" with respect to the uncertainty principle. –  mbeckish Jun 13 at 13:22
    
@mbeckish : The characteric size of an object is a general concept, not especially linked to quantum mechanics. You may see the characteric size of an object as a mean of the $3$ spatial dimensions of this object. –  Trimok Jun 14 at 10:40

I'm going to make an attempt. Your formulation suggests contradictions with relativity but since I don't know relativistic quantum mechanics on one hand, and on the other the problems you detect may not be relativistic in an essential way, I will just consider the Schrödinger picture.

First of all, it is certainly possible to interact with a particle that has a non-zero probability of being measured at arbitrarily large distances, as long as it also has a non-zero probability of being here.

That is not really the issue though, since you ask about the state after the measurement. Before your measurement your particle is in a state that obeys the uncertainty principle. In fact, a refinement of this principle will show that if the possible values for either one of position and momentum are bounded, the possible values for the other have to extend to infinity.

When you measure the momentum of the particle, the state of the particle collapses to the most general state that is compatible with your measurement. This will not be the same state as before the measurement, unless it was already in that state. By measuring the momentum with arbitrary high precision, indeed you end up with arbitrary high uncertainty in the position. In the past the explanation was that the more precisely you want to measure one property, the more you'll have to disturb the other, but I think nowadays that view is abandoned as not useful. It does follow logically though from the measurement principle stated above. Strange as it is, this seems to be how nature behaves.

What if the uncertainty extended beyond the bounds of the universe?

If by the universe you mean a bounded space to which the particle is strictly constrained (by considering the potential to be infinity outside, i.e. we got a particle in a box) then the uncertainty principle implies that a state with definite momentum is not a possible outcome, since that would indeed imply a state in which there is a non-zero possibility to find the particle outside the universe.

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I don't believe a universe of finite size is the same as confinement in a box. Consider a one dimensional example. Confinement in a box means that there is a potential to your left and to your right, preventing you from going further in either direction. A finite sized (closed) universe means that your one dimension is a loop. There is no potential confining your space - it's just that space is closed, so if you go far enough, you end up back where you started. –  mbeckish Jun 13 at 1:38
    
"First of all, it is certainly possible to interact with a particle that has a non-zero probability of being measured at arbitrarily large distances". Even though we don't have a quantum theory of relativity, I still think you can't make a claim in quantum mechanics that contradicts a claim in relativity, without admitting that one of them must be wrong. Since information can't travel arbitrarily large distances in a short amount of time, I don't see how you can get information about the momentum of a particle that is a very large distance away. –  mbeckish Jun 13 at 1:46
    
Unless the time is also uncertain - maybe we're measuring the momentum of a particle one light-year away as it was one year ago? –  mbeckish Jun 13 at 1:46
    
@mbeckish if you want to consider a closed universe there is no problem, because the uncertainty principle in that form is specifically for $\mathbb R^n$ –  doetoe Jun 13 at 1:56
    
@mbeckish about the relativity: This is special relativity. There is a good theory of special relativistic quantum theory, just that I'm not very familiar with it. Nevertheless I will admit that quantum mechanics and general relativity cannot both be correct in their present form: this is generally accepted. Finally in this case there is no contradiction: no information has to travel –  doetoe Jun 13 at 2:04

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