What is the wave function for a single electron?

[Quandry]

If you know where to look for an electron (e.g. in an atom or an experimental setup) it is quite understandable that, until you know exactly where it is, there is a calculable probability of where it might be. However, if we take the case of an un-associated electron in space, it would seem that the probability of it being somewhere is 1 and a probability of it being nowhere is 0. Which is a crass way of saying that there is no way to determine a probability and therefore it has no wave-function.
A somewhat philosophical question, but to extend it, if you fire a single electron in a slot experiment, what makes it take up a position in the interference pattern, and how do you know that it has done that? And if you fire multiple single electrons which create the interference pattern, what interference has there been?

 

[BvU]

Which is a crass way of saying that there is no way to determine a probability and therefore it has no wave-function.

Not correct and the logic is unfounded

what makes it take up a position in the interference pattern

chance does

how do you know that it has done that

If you place a detector there it gives a signal

what interference has there been

the same interference as for a single electron

Perhaps you want to follow some of the Feynman lectures on this subject ?

 

[Nugatory]

However, if we take the case of an un-associated electron in space, it would seem that the probability of it being somewhere is 1 and a probability of it being nowhere is 0. Which is a crass way of saying that there is no way to determine a probability and therefore it has no wave-function.

That does not follow. The condition that there is a 100% chance that the electron is somewhere (and therefore a 0% chance that it is nowhere) just requires that the wave function $\psi$ has the property that $\int\psi^*\psi$ over all space is equal to 1. It's easy to find wave functions that satisfy Schrödinger's equation for a free particle and that have this property.

 

[Nugatory]

A somewhat philosophical question, but to extend it, if you fire a single electron in a slot experiment, what makes it take up a position in the interference pattern, and how do you know that it has done that?

It makes a dot on the screen where it hits.

And if you fire multiple single electrons which create the interference pattern, what interference has there been?

Each individual electron interferes with itself.

 

[Quandry]

Not correct and the logic is unfounded
chance does
If you place a detector there it gives a signal
the same interference as for a single electron

Perhaps you want to follow some of the Feynman lectures on this subject ?

Hmmm! Helpful.
I have followed the Feynman lectures.

 

[Quandry]

It makes a dot on the screen where it hits.

My question was really how do you know the dot is part of an interference pattern. But no worries.

Each individual electron interferes with itself.

Which means that the electron has the attributes of a wave (no surprises there) and passes through both slots with a phase relationship which results in energy interference. The interference pattern is dependent on the wave characteristics, slot characteristics etc. which result in energy peaks and troughs.
However, with the electron slot experiment it seems that the energy resolves to a single point. I cannot find an explanation for this.

 

[Nugatory]

My question was really how do you know the dot is part of an interference pattern.

We don't, when we're looking at a single dot from a single electron. We cannot learn anything about the different probabilities of electrons landing at different points on the screen by running the experiment with a single electron; we need many observations of many electrons for that and this is no different than ordinary classical probability.

For example, suppose that I tell you that I have an honest coin that will come up heads half the time it is thrown and tails the other half of the time, and you want to run an experiment to see if I'm telling the truth. If you throw the coin once and you get heads, you have no way of knowing whether that's because the coin is weighted to come up heads more often, or whether it is an honest coin that just happened to come up heads this time, or whether it is weighted to come up tails most of the time and you just got lucky on this throw. But if you throw the coin 1000 times and you get 753 heads and 247 tails... You can be certain that the coin is weighted to favor heads. Detecting interference effects works the same way; when many dots appear in one area of the screen and few in another we know that the probability of a dot appearing in some regions is higher than in others.

Which means that the electron has the attributes of a wave (no surprises there) and passes through both slots with a phase relationship which results in energy interference. The interference pattern is dependent on the wave characteristics, slot characteristics etc. which result in energy peaks and troughs.
However, with the electron slot experiment it seems that the energy resolves to a single point. I cannot find an explanation for this.

That's an experimental fact about how quantum particles behave (and it's a historical accident that we even use the word "particle" - this behavior is unlike anything that you'd expect from the ordinary English-language meaning of the word). When they interact with matter they deliver their energy and momentum at a single point, and you have to do the wave calculation to find the probabilities of which point it will be.

 

[Quandry]

We don't, when we're looking at a single dot from a single electron. We cannot learn anything about the different probabilities of electrons landing at different points on the screen by running the experiment with a single electron; we need many observations of many electrons for that and this is no different than ordinary classical probability.

I understand that we cannot learn about probabilities from a single sample. But it is not the probability of where the 'dot' is that I am trying to understand, it is why is it a dot?.

That's an experimental fact about how quantum particles behave (and it's a historical accident that we even use the word "particle" - this behavior is unlike anything that you'd expect from the ordinary English-language meaning of the word). When they interact with matter they deliver their energy and momentum at a single point, and you have to do the wave calculation to find the probabilities of which point it will be.

Trust me, I do not perceive electrons as buckyballs flying around. My background is electronics and telecommunications, so waves are, you might say, are my bread and butter. You have what is effectively a plane wave arriving from a distance passing through two slots and effectively becoming new wave sources. The two wave sources interact with each other and an interference pattern is created, with the strongest constructive interference, and therefore highest energy, between the two slots. but there are always other, lesser, constructive energy peaks. The single 'dot' of the electron implies that there is a single constructive interference peak when an electron 'interferes with itself'.
So, to create the traditional interference pattern for a series of individual electrons interfering with each other, there is an implication that each electron, when it arrives at the slots, 'knows' about the previous ones and acts accordingly, to create the pattern. I don't think so.
So, if an electron looked like a particle when it arrived at the slots, it would not interfere with itself. If it looked like a wave, it would, but the result could not be (?) a single point of energy.
If a series of electrons arrive as waves which somehow create interference patters with single points of energy, cumulatively create a typical interference pattern which conform with probability maths - what influences the probability - why is one electron more likely to be 'heads' than 'tails'?

 

[Nugatory]

The single 'dot' of the electron implies that there is a single constructive interference peak when an electron 'interferes with itself'.

The point where the dot appears does not imply a single constructive interference peak. When an electron move through the apparatus we calculate the interference pattern for its wave passing through the two slits, and that gives us the probability of the electron landing at any given point on the screen. That probability is highest at the constructive interference peaks, lowest at the destructive interference troughs, and in between everywhere else. When the next electron enters the apparatus, we again calculate the interference pattern for that one's wave, and of course because the conditions are the same we get the same result with the same pattern of high probabilities at some points, low probabilities at other points, and in between at the in between points.

Do this with enough electrons and there will be more dots in the areas where the probability was high than in the areas where it was low, so the visible interference pattern will emerge. This happens without any electron 'knowing" about where the others landed; each one is acting according to the probabilities calculated from its wave without involving any of the others.

 

[Quandry]

All good, thanks. But when you say 'electron landing', we are (in a single electron situation) converting energy in interference waves into a single point of energy (experiments show a single dot for a single electron through the double slot apparatus). This point is not probabilistic, it is dictated by the interference pattern generated at the two slots. Probability comes into it when subsequent electron arrive and there is a probability factor in the subsequent electrons generating the same interference pattern.

Unless, of course, we say that the energy of a wave is contained in a single point (a particle) which pops out at the screen. However, this is inconsistent with interference waves where the nodes and nulls are created by interaction of the energy in the wave. If there is only one point of energy, as implied, there is no energy spread to create interference.

 

[Nugatory]

This point is not probabilistic, it is dictated by the interference pattern generated at the two slots.

Once the dot has appeared, there's no doubt that it appeared at this spot and not that spot, but which point it lands at most certainly is probabilistic. That's one of the axioms of quantum mechanics, and if you google for "Born rule" you'll find some decent explanations of how it works.

 

[Quandry]

Went back and did a review of the Born Rule, rewatched Feynman's lecture on the topic, reviewed Young and a few others I have collected over time. Could not find the source of the 'oft quoted' statement by Dirac that electrons interfere with themselves.
However, I fully understand and accept that repeat single electron experiments result in the interference pattern and I am sure that an electron arriving at the double slots as a wave will conform with Huygen's principle and generate two waves which will interfere with each other. But I do not understand how this results in a single point of energy.
I am going around in circles, need to contemplate more.
Thanks for your perseverance.