Is there a theoretical or technological limit for arbitrary precise measurement of the position of a point particle in QM?

Question

I have read this question:

Where pglpm says:

So the answer to your question is that in a single measurement instance we actually can (and do!) measure position and momentum simultaneously and with arbitrary precision.

and where mmesser314 says:

The electron doesn't have a precise position.

As far as I understand, the electron is defined as a point particle in the SM, with no spatial extension or size. One of the answers says we can measure position with arbitrary precision. The other one says the electron does not have such a precise position, just a probability distribution. But that is not what I am asking about. Just to clarify, I am not asking about the HUP or any kind of simultaneous measurement of observables. What I am asking about is when they both talk about arbitrary precise position for a point particle.

The electron is a point particle, that is, defined as the smallest thing we know of. Smaller then any measuring device. And spacetime is currently known to be continuous, any distance (for position measurement) can be divided into yet smaller and smaller pieces.

For a point particle, that does not have spatial extension, what do we mean when we say arbitrary precise measurement of position? Can this be farther improved with new technology or not?

As far as I understand, the precision is limited by cross sections. This is the limit of precision, because to measure the observable of a QM particle, we need to interact with it, with another QM particle. These have a cross section, and as I understand that limits the precision. Can this then be farther improved somehow?

Is this limit set by the QM nature of QM particles, and their cross section, or can this be improved using better technologies?

So basically, is the limit theoretical, or technological?

Question:

1. Is there a theoretical or technological limit for arbitrary precise measurement of the position of a point particle in QM?

Answer 1

Yes, though not within simple, non-relativistic quantum mechanics. Instead, the limit comes from relativistic quantum field theory (RQFT), and it is given by the Compton wavelength: the maximum resolution of position for a particle of mass $m$ is on the order of the length scale

$$\lambda_{\mathrm{Compton}} := \frac{\hbar}{mc}$$

For the electron, this is on the order of 0.39 pm. By comparison, a typical atom is around 200 pm in diameter. It can thus be considered as a stronger version of the Heisenberg uncertainty principle that appears in the relativistic context.

Answer 2

If you measure something with infinite precision it means that you have another variable with infinite uncentarty, it means that in principle it can end up being infinitely large, for example if we have $$\Delta x=0$$ it implies $$\Delta k=+\infty$$ so the average Energy $$\langle E\rangle =+\infty$$

So you need infinite energy, if you don't want your measurement to form a black hole you must use a limited amount of energy. So yes, there are limits on the precision of measurements.

Other examples are the fact that $$\Delta x<\textrm{size of the universe}$$ and $$\Delta t< \textrm{the time left before the thermodinamic death of the universe}$$

Answer 3

Is there a theoretical or technological limit for arbitrary precise measurement of the position of a point particle in QM?

One has to keep clear what technology is and what theory is.

Quantum mechanics is a very successful physics theory that presently predicts the behavior of nature below the nanometer scale. As a theory it has the HUP within its mathematics, the commutators of pairs of variables. Within this framework , even though the elementary particles are axiomatically point particles, there is a limit on the precise measurement of the position of an electon, for example, inherent in QM, which is a probabilistic theory.

There are deterministic theories, from which they expect the probabilistic nature of QM to emerge. In such theories the HUP will be emergent and position will be equally indeterminate within HUP for the particles of the standard model. If new predictions can be made for evading the HUP, or HUP like constraints is not known to me.

Our technology at the micro level is based on the theory of QM, and so, even if one could think of ways to reduce measurement errors to tiny values, one could not escape the probabilistic nature that induces the HUP.

Only if one could design an experiment that would invalidate the HUP, i.e. show that nature is more than quantum mechanics, one might be able, given better technology, to show that only measurement errors limit the precision of measurements.