B Heisenberg Uncertainty Principle question

[skeptic2]

TL;DR

Does the Heisenberg Uncertainty principle say that if the position of an entangled particle is measured, the momentum of the other entangled particle cannot be simultaneously measured?

See above summary.

 

[PeterDonis]

Does the Heisenberg Uncertainty principle say that if the position of an entangled particle is measured, the momentum of the other entangled particle cannot be simultaneously measured?

No.

 

[DaveC426913]

You are encouraged to do your own research, but I can;t resist a hint:

The product of the precision of the two measurements is a constant. In other words, the more precisely you measure one, the less precisely you can measure the other.

 

[Nugatory]

Does the Heisenberg Uncertainty principle say that if the position of an entangled particle is measured, the momentum of the other entangled particle cannot be simultaneously measured?

No, the HUP does not limit the accuracy of measurements. We can measure the momentum or position of either particle in the entangled pair to as many decimal places as our measuring device will allow, and there's no limit on how many decimal places that might be as we design ever better measuring devices.

Instead,the HUP is a constraint on state preparation. We set up our experiment so that we create an large number of entangled pairs and every time we measure the position of particle A we get the same result, within the limits of of out measuring device. We will find that momentum measurements of particle B (and A, which is why can talk about the HUP in single-particle cases) every time will be different, and the spread of the differences is consistent with the HUP.

Free advice: When you're reasoning about entangled particles it's generally easier to take David Bohm's advice and consider spin-entangled electrons or polarization-entangled photons, instead of position/momentum entangles pairs.

 

[DaveC426913]

No, the HUP does not limit the accuracy of measurements. We can measure the momentum or position of either particle in the entangled pair to as many decimal places as our measuring device will allow, and there's no limit on how many decimal places that might be as we design ever better measuring devices.

My post #3 is wrong?

 

[PeroK]

My post #3 is wrong?

Very wrong!

 

[PeroK]

You are encouraged to do your own research, but I can;t resist a hint:

The product of the precision of the two measurements is a constant. In other words, the more precisely you measure one, the less precisely you can measure the other.

The HUP says nothing about the precision with you can measure a quantity. That's depends on your experimental apparatus.

Instead, the HUP is about the variance in measurements (on an ensemble of identically prepared systems). Any variance due to the precision of your measuring apparatus is additional to the HUP.

 

[martinbn]

My post #3 is wrong?

Don't worry, most get it wrong at first. For some reason many textbooks do not explain it very well, and most popular books state it wrong.

 

[weirdoguy]

and most popular books state it wrong.

Even professor at my uni was wrong about it (second year of physics, lectures on atomic physics and general introduction to QM).

 

[pines-demon]

The confusion comes from the principle of complementarity by Bohr. The problem is that Bohr was so philosophical about it that it is impossible understand it even if you read historical takes of it. He mixes a lot of old quantum theory and wave-particle nonsense.

The modern version is to say that there are non-compatible bases, due to non-commuting operators, like position and momentum $[\hat x , \hat p]=\hat x \hat p - \hat p \hat x=i\hbar$. For non-commuting operators you cannot find a common basis. If you have an eigenstate of one operator (maybe because you measure it), you cannot have an eigenstate of the non-commuting operator. This distinction does enter into Heisenberg's uncertainty principle but I would say it is more fundamental than that.

Now in entangled states à la EPR paradox, yes, measuring the position of the first particle fixes the position of the second particle, and thus momentum of the two particle is now in a superposition of many eigenstates of momentum. You can choose to measure the momentum of the second particle, after measuring the position of the first particle, but in that case the position and momentum will be maximally uncorrelated, meaning that you cannot infer anything about it. Even if you perform more than one run of the experiment, you cannot even tell if the particles are entangled, you need to perform measurements in many different bases.