Can probability waves be "focused"?

[Grinkle]

TL;DR

There is no model for "collapse". Is there any wave function evolution that can be interpreted as focus?

I really cannot ask this question well. I can only hope its not simply a waste of the readers time. I won't finish every sentance with "maybe I'm wrong", just assume its in my mind every time I hit the period key.

An electron on a screen leaves a pixel spot, this pixel spot is a measurement of the position of the electron. Regardless of the screen technology, I expect the diameter of the pixel spot to be many orders of magnitude larger than however one may choose to define the diameter of an electron. The location of the electron has not been measured very precisely relative to the limits of uncertainty by this process.

This makes wonder if there is any probability waveform evolution of a small enough number of particles to analyze that shows a focusing (as opposed to a collapse) where the probability of an electron being detected within some radius becomes zero outside of some finite diameter and all of the probability is inside some finite diameter.

Or is this just another way of talking about "collapse"?

 

[PeroK]

Summary:: There is no model for "collapse". Is there any wave function evolution that can be interpreted as focus?

The single slit is an attempt to focus a beam of particles or light. It works well up to a point, but when the slit becomes too narrow the beam diffracts, rather than being focused any further.

 

[Grinkle]

The single slit is an attempt to focus a beam of particles or light.

Thanks for the response - I assume you are talking about an experiment, not an analysis, and from there I assume there is no analytical path to getting all 100% of (for example) position probability within a finite diameter. Do you know if that is true?

I think I am asking more generally about one of a pair of joint-operators, but I am trying to stay away from words with that many syllables and hyphens.

 

[PeroK]

Thanks for the response - I assume you are talking about an experiment, not an analysis, and from there I assume there is no analytical path to getting all 100% of (for example) position probability within a finite diameter. Do you know if that is true?

Theoretically, you can have a particle in an infinite potential well. In practice, any potential well is finite, so there is always a small probability of the particle leaking out.

There's also a Penning trap:

https://en.wikipedia.org/wiki/Penning_trap

 

[hutchphd]

Summary:: There is no model for "collapse". Is there any wave function evolution that can be interpreted as focus?

I expect the diameter of the pixel spot to be many orders of magnitude larger than however one may choose to define the diameter of an electron

The electron is a "point particle" and so there is no particular content to this statement.
The electron microscope uses "electron lenses" to focus electrons. The reason the images are so much higher resolution than a light microscope is that for a given energy the wavelength of an electron is much shorter than that of the same energy photon. Or conversely the energy of a 1 Angstrom electron is 1.5 eV while that of a 1 Angstrom photon is 12,300 eV .
So what do you mean by focus ? The issue of localization for a massive particle is complicated and that for a photon is differently complicated. You need to carefully define your questions or you will be in perpetual confusion

/

 

[Grinkle]

The electron is a "point particle" and so there is no particular content to this statement.

My thinking here is that when a measurement of the electron position is taken by the screen, this is used as an example of waveform collapse. The position of the electron has been measured to be 100% at one spot and 0% everywhere else. I don't think a dot on a screen is really one spot, its a circle (more or less) with a finite diameter that still contains lots of possible spots. This made me wonder if the probability wave itself can analytically evolve to show the particle (it doesn't matter for my question whether its an electron or something else that is behaving in a quantum manner) position probability being 100% contained in a circle or sphere or boundary that is less than infinite.

I am not talking about focusing electrons per se, I am asking about probability wave functions and whether these probability distribution wave functions can be "focused". I use the word "focus" to illustrate the picture in my head of a position that can be anywhere in the universe evolving to a position that can only be in some bounded part of the universe.

I think the answer to my question is no.

 

[hutchphd]

A beam of electrons can be sent through an aperture as mentioned by @PeroK. Also an electron in an atom is localized.
So without using the word "collapse", what ( in plain english) is the question you are trying to answer?

 

[Grinkle]

what ( in plain english) is the question you are trying to answer?

A probability wave snapshot at a given time is a probability distribution of finding a particle at any arbitrary position in the universe, and the sum of all probability is 1.

Is it possible to do some analytical combination of probability wave 1 and probability wave 2 that after letting the combined wave evolve over time results in probability wave 3 that has only a finite subset of the universe as the possible positions for the particle in either wave 1 or wave 2 or both?As a side question / check on an assumption I am making -

Also an electron in an atom is localized.

I think that quantum mechanics says an electron in an atom has a non-zero chance of being observed anywhere in the universe. Am I wrong?

 

[CoolMint]

I think that quantum mechanics says an electron in an atom has a non-zero chance of being observed anywhere in the universe. Am I wrong?

I think hutchphd means that a bound electron in an atom is much more localized than a free electron.

 

[hutchphd]

To contain an electron absolutely requires infinitely strong barriers. Why is this important? In the real world such barriers can only be approximated and then the probabilities made small. De minimus may not be zero but why do you care? Science deals ultimately with observation.
In a hypothetical universe we get to choose lots of things that are fictional.

 

[Grinkle]

Why is this important?

I was not intending to ask specifically about electrons, I was clumsily asking about probability waves. I was referring to electrons because I was thinking about the double slit experiment to try and collect my thoughts while posting my question. There is no practical consequence that I can come up with regarding whether an electron will ever be observed in a totally unexpected spot. Its not important.

In a hypothetical universe we get to choose lots of things that are fictional.

Agreed. I was hoping there is a known analytical path to a zero probability in the wave function of something being arbitrarily far away from where it is expected to be using non-QM analysis to set ones expectation. There was some discussion in a different thread about a wave function having a non-zero value which started me wondering if we have any examples of wave functions with zero values somewhere and non-zero values in other places.

 

[PeterDonis]

was hoping there is a known analytical path to a zero probability in the wave function of something being arbitrarily far away

There is: make the potential infinite outside a finite distance from the center. The issue with that is, as others have pointed out, that there is no such thing in the real world as an infinite potential. So what it amounts to is that we have a choice of analytical models: we can use an infinite potential to get zero probability outside a finite range, and accept that we are modeling an idealized potential that is not physically realizable; or we can use a finite potential to reflect what kind of confinement we can physically realize, and accept that we are modeling an idealized wave function that is nonzero out to infinity even though we aren't ever actually going to measure a confined quantum system to be at an arbitrarily large distance away. There is no known analytical model that exactly represents both things.

 

[hutchphd]

There are many parts of quantum mechanics which do not comport with "normal" everyday experiance. Quantum "tunneling" is but one of them. I suggest being mollified by R P Feynman's attitude that we don't have to like it but it seems to be true.

 

[Klystron]

Summary:: There is no model for "collapse". Is there any wave function evolution that can be interpreted as focus?
...
An electron on a screen leaves a pixel spot, this pixel spot is a measurement of the position of the electron. Regardless of the screen technology, I expect the diameter of the pixel spot to be many orders of magnitude larger than however one may choose to define the diameter of an electron. The location of the electron has not been measured very precisely relative to the limits of uncertainty by this process.
...

Consider employing a statistical model different from single pixel activation on a screen, such as bunching of electrons in a traveling wave tube (TWT), or perhaps a simpler electronic model such as a vacuum tube triode. I have found visualizing a statistically significant bunch of electrons more useful to understanding and teaching wave mechanics than attempting to isolate an individual electron.

If you wish to continue using your pixelated screen model, consider examining the electron beam in reference to understanding wave function in preference to tracing the path of an individual particle. Just a suggestion, not criticism.

 

[StevieTNZ]

I think that quantum mechanics says an electron in an atom has a non-zero chance of being observed anywhere in the universe. Am I wrong?

A few authors, such as Brian Cox and Jeff Forshaw, in their book 'The Quantum Universe' make that claim. Also Stephen Hawking's in 'The Grand Design' (more in relation to the double slit experiment). Both are pop-sci references, but I suspect someone can provide a textbook in which the same is said.

I think in 'Quantum Mechanics: A New Introduction' by Kenichi Konishi and Giampiero Paffuti has some details on it.