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
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!
No comments:
Post a Comment