Measurement problem - will this work?

[Gary Venter]

TL;DR

The uncertainty principle is usually stated epistemologically: if you know more about position you must know less about momentum. But mathematically it also looks like the opposite works too: if the momentum uncertainty increases, then the position uncertainty decreases. Is this a possible path to resolving the measurement problem?

Consider the measurement problem for an electron in a single-slit experiment done one electron at a time. There are two interlinked questions, but some consider just one of them to be the actual measurement problem. The first question is concentration - a fairly uncertain distribution condenses to a much more localized spot, but there is nothing apparent in the device that would concentrate the wave function. The other is randomness - how does that specific spot result? The many-worlds approach gives one possible answer to the randomness question but does not address the concentration issue - so some take that issue as the measurement problem.

One complicating factor is the idea that measurement changes a wave into a point particle. This idea has a long history but is extraneous in this context. No measurement available can distinguish a point particle from a very concentrated wave function - say 10^-40 cm across. Taking out the particle idea still leaves both questions - how does the wave function get condensed and how is it's specific location region determined? But at least then the electron has an ongoing wave function.

Now the momentum. The epistemological expression of the uncertainty principle says that when we know more about the location of the electron we know less about its momentum. In terms of wave functions, that means the location function has become more concentrated and the momentum function more dispersed. Mathematically these two wave functions are Fourier transforms of each other. Thus when one of them changes, the other one also changes. I once saw that fact expressed as there being only one actual object in the quantum field and position and momentum are views of that object using frameworks that are perpendicular to each other. I don't know what that means and maybe said it wrong but in any case, the two move simultaneously.

This would suggest that the mysterious concentration of the position could have been brought about by dispersal of the momentum. You wouldn't get that purely epistemologically, because knowing less about something would not imply that you know more about something else. There are all kinds of ways you could get less knowledgeable about the position - like poor tracking, or the machine going off, ... . But the wave functions do move simultaneously, and neither is required to be the driver of the other.

One problem with this is that widening uncertainty of a wave function might not in itself narrow its Fourier transform. The widening might have to happen in specific ways - maybe not being restricted from certain directions or something like that. So the math would have to be looked at more closely. Still, in this experiment the spread of the momentum function does correspond with a great concentration of the location function, so whatever any other conditions might need to be fulfilled, they appear to be.

The measuring device looks like it would spread the momentum. Coming from the slit, the direction of the particle is not too widely dispersed. As it gets closer to the device, the electron must interact electrically with all the electrons in the device in different ways, which could spread the direction component of the momentum. The Pauli exclusion principle could even create restrictions in where the electron might be headed - maybe. In any case, the device does look like it would disperse the momentum of the electron, which is one result of the experiment, and that might be the key to how the location gets focused.

That's the basic idea. it would have to be expanded upon mathematically, but it seems to be a potential avenue for working on the position-concentration problem.

 

[DrClaude]

But mathematically it also looks like the opposite works too: if the momentum uncertainty increases, then the position uncertainty decreases.

That is incorrect. There is no particular limit to maximizing the uncertainty: we can create states where both $\Delta x$ and $\Delta p$ are as big as we want.

Your post is very wordy, and we don't do physics with words, but with math. Your post sounds like you have learned about quantum mechanics from popularizations, not from actual textbooks.

The generalized uncertainty principle is
$$
\sigma_A \sigma_B \geq \frac{1}{2i} \braket{[\hat{A}, \hat{B}]}
$$
and, with apologies for being blunt, if you do not understand that equation, you will have to learn quantum mechanics at least to that level (and most probably way beyond) before you can really get a grasp of the measurement problem (which is not due to the uncertainty principle).

 

[PeroK]

TL;DR Summary: The uncertainty principle is usually stated epistemologically: if you know more about position you must know less about momentum. But mathematically it also looks like the opposite works too: if the momentum uncertainty increases, then the position uncertainty decreases. Is this a possible path to resolving the measurement problem?

I looked at your introductory post and you are perhaps more interested in philosophy than physics. As far as these forums are concerned, we are interested in physics as an academic discipline. And that means ideas are backed up by mathematics.

This post is a collection of vague ideas that have little relevance to QM as a scientific subject. And, therefore, there is nothing to discuss.

If you are serious about learning QM, you should start with an undergraduate textbook. I would recommend Griffiths, but there are lots of alternatives and several threads on here discussing the merits of each.

I have the second edition of Griffiths, which you can pick up for very little now the third edition is out.

https://www.abebooks.co.uk/servlet/...NI5zI8h0EayFO6FOWqK3H8XMmgkIhTFBoC3LkQAvD_BwE

 

[PeterDonis]

The uncertainty principle is usually stated epistemologically

A claim like this needs to be backed up with references.

the idea that measurement changes a wave into a point particle. This idea has a long history

Same comment here about references.

That's the basic idea. it would have to be expanded upon mathematically, but it seems to be a potential avenue for working on the position-concentration problem.

This looks like personal speculation, which is off limits here. Do you have a reference for this idea?

 

[jk22]

For a qubit the measurement could be written as a transition operator $\Psi_{n+1}=T\Psi_n/||T\Psi_n||$ where T is not forcedly a unitary operator.

But the problem is that it remains probabilities and in fact there would be infinite number of ways to extrapolate. It is like saying something imaginatively about unknowable thing.