Uncertainty principle - momentum so precise that uncertainty of position is outside light-cone?

Question

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?

Answer 1

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!

Answer 2

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 $d$.

Answer 3

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.