Interpretation of the double slit experiment and Schrödinger's wave function?

Hi I'm trying to understand the double slit experiment after having studied a bit of quantum mechanics and I'm having a bit of trouble putting the two together. So if we fire electrons one by one through two slits we get an interference pattern so they are acting like a wave just as if we were shining light for example. I don't know what the Schrödinger's wave function looks like for representing this electrons but I imagine there would be something like the state of having gone through one of the slits with some probability and another state with the other slit. There are a couple things that I don't understand if we represent the electrons in this way:

1. I know it's common to say that when states are in superposition the electron is in all of them at the same time but I thought that was a vague way of explaining it when in reality what's going on is that there is a probability for each state and we don't know which one it is until we measure (so we say it's all of them). If we go with the probability understanding of the wave function it would mean that the electron went through one of the slits "randomly" and we don't know which until we measure. But even if we don't measure it it still went through just one of those slits so why is there an interference pattern? What is interfering with the electron?

2. Also even if the reality was that the electron really did go through both slits at the same time, how is this possible? And if the wave function collapse when measured wouldn't the fact of looking at the screen to see the interference pattern collapse the wave functions of the electrons and make the interference pattern disappear. Why does it only disappear when we specifically detect through which slit the electrons are going through? Why doesn't looking at the screen count as an observation/measurement?

• When you accelerate electrons through a slit experiment, they emit billions of photons. These photons disperse in front of the electron and reflect back, with millions of them colliding with the electron as it progresses towards the detection screen. The electron persists in emitting photons throughout its journey. The resultant interference pattern created by these photons interacts with the electrons, effectively guiding them into a unified pattern. Jul 7, 2023 at 17:39
• Re, "... they are acting like a wave." Maybe more accurate to say, they are acting in a way that seems to be related to a wave. Each electron interacts with a single atom of the detector screen. The spatial distribution of those interactions is predicted by the way in which an imaginary* wave would diffract through the slits and interfere with itself on the other side. *I say, "imaginary" because I can imagine it. What I cannot imagine—maybe only because I am not a physicist—is, what exactly is waving? Jul 7, 2023 at 18:55
• The EM field guides electrons and photons. Electrons and photons always have wave properties due to the EM field .... the wave function can change dynamically .... the EM field is very dynamic ..... see my comments below. Jul 9, 2023 at 12:54

And if the wave function collapse when measured wouldn't the fact of looking at the screen to see the interference pattern collapse the wave functions of the electrons and make the interference pattern disappear. Why does it only disappear when we specifically detect through which slit the electrons are going through? Why doesn't looking at the screen count as an observation/measurement?

In collapse interpretations, when the wavefunction is observed it collapses to a single localised peak (like a particle), but then thereafter starts spreading out like a wave again.

So if you observe the wavefunction as it passes through the slits, it collapses to a point passing through one slit, then this single-peak wavefunction spreads out again as it heads for the screen. At the screen it is observed again, collapsing to one point on the screen, which has the distribution of a single-slit spread (i.e. no double-slit interference pattern).

If you don't collapse the wavefunction at the slits, the wave passes through both slits at once, and being a linear differential equation, the two-slit pattern is the sum of two one-slit patterns, which results in an interference pattern. At the screen, the wavefunction with the two-slit interference pattern collapses to a single point.

Looking at the screen does count as an observation/measurement in both cases. But the wavefunctions being collapsed are different - either the one-slit pattern if you have previously collapsed it by observing it at the slits, or the two-slit pattern if you haven't.

I know it's common to say that when states are in superposition the electron is in all of them at the same time but I thought that was a vague way of explaining it when in reality what's going on is that there is a probability for each state and we don't know which one it is until we measure (so we say it's all of them).

It is a vague way of explaining what is going on, but rather than being a case of a particle of uncertain position that we are incorrectly/fuzzily treating as a probability wave, it is actually a wave that we are incorrectly/fuzzily treating as a particle being in many places at once.

In the collapse interpretations, it is not a particle until you observe it. It is not a probability, either, until you come to actually observe it. Probabilities are real numbers between 0 and 1. Probabilities aren't complex numbers which can destructively interfere with one another. The wavefunction behaves as a complex wave, which when observed (and only then), gives rise to an associated probability of observing it at each point.

As you say, the single electron must go through both slits at once, or else there would be no interference. An electron passing through only one slit would have no way of knowing about the other; whether it was open or closed. Waves can do this. Particles can't.

The tricky part is interpreting what happens when we observe it. The wavefunction hits the entire screen at once, and interacts (lights up) at only one place on the screen. The rest of the screen somehow knows not to respond, even if there is no way for a signal to get there from the one place in time. Collapse is non-local - the transition from the spread-out wavefunction to the single-peaked, highly localised wavefunction appears to propagate faster than light, and hence, in a relativistic universe, backwards in time for some moving observers. It would be understandable if you rejected this option as logically impossible too.

There is an alternative interpretation that retains locality, which requires you to treat the observer as a wavefunction too. Then when the observer interacts with the electron, the wavefunctions become 'correlated'. Just as the spread-out electron is a sum of localised electrons at each point of the screen, so the observer becomes a sum of observers, each observing the electron at a different point. Just as different parts of the electron's wavefunction don't interact with one another (there is no electromagnetic repulsion between different parts of the wavefunction, for example), so the summed observers don't interact with one another. The physics is deterministic (all outcomes happen, every time) and local (the observers seeing it hit one part of the screen can never perceive any observers seeing it hit another part, even if they all go to the same place later). It only appears random and non-local because we can only see our own small part of the bigger picture. There is no collapse, only correlation. There are no particles, only waves. One wave looks like a particle to another wave when they are perfectly correlated; when they move in sync. Not everybody likes that idea, though.

• Thanks for the answer! For your last answer (the alternative interpretation) , are you talking about the many-worlds theory? Because it sounded kind of like that but you didn't say it explicitly. Jul 8, 2023 at 8:26
• @Omeglac You should also consider that the EM field is very dynamic and everywhere in space .... and most importantly that the electron is interacting with this EM field while it is excited in the electrode ... long before it is emitted. Thus the EM field is what goes thru the 2 slits ..... has makes a "virtual" pattern on the screen .... the electron follows the most probable paths and are observed to "interfere". Jul 9, 2023 at 12:34

If we go with the probability understanding of the wave function it would mean that the electron went through one of the slits "randomly" and we don't know which until we measure.

You have to understand that Probabilities in Quantum Mechanics are fundamentally different from probabilities in statistics. Imagine rolling a dice. You watch the dice roll along the table. If you looked hard enough and measured it's exact movement, you could predict on which side the dice is going to land. Meaning, the information about where the dice is going to land is already there before it stops rolling. You're just ignorant about it. And to help you with your ignorance, you use statistics. You say, it's going to be a 6 with a probability of 1/6.

The point of Quantum Mechanics is that (at least in the Kopenhagen Interpretation), the information about the exact state of the system (for example the position of the electron) is really not there before you measure it. So it's not that the electron really did go through one of the slits, and you just didn't measure it; the information about the position really didn't exist.

If you let the electron through the double slit without observing it, it really goes through both slits. But if you measure it, you force it into a certain state (i.e. electron went through the left/right slit). This collapse of the wave function is a very weird process, that we still don't really understand. It's called the measurement problem.

And if the wave function collapse when measured wouldn't the fact of looking at the screen to see the interference pattern collapse the wave functions of the electrons and make the interference pattern disappear.

No, because it matters at which point you do the measurement. If you do the measurement (collapse the wave function) at the slit, the wave won't have a chance to interfere with itself, and you get ordinary particle behavior. If you only measure the electrons at the screen, you give the wave function the opportunity to interfere with itself, and only after the interference, you force the electrons into a certain position.

• Thanks for your answer! One doubt, you say that in QM the information about the position of the Electron doesn't really exist until we measure it. What does Schrodinger's wave function show then? I understood it (vague explanation) as all the possible positions of the electrons with their respective probabilities before measuring it and forcing it to collapse to a certain position (assuming we express it in the position basis). Isn't this already giving some information about the position of the Electron before measuring it? How can this information not exist beforehand then? Jul 8, 2023 at 8:18
• @Omeglac Well, this kinda cuts to the heart of wave-particle duality and the question "what is an electron?" As long as it's unperturbed, the electron REALLY IS the wave function. At that point asking "where exactly is the electron" is like asking for the precise position of a wave in the ocean. The question doesn't make sense; it's in many places at once. We can interpret the wave function as a probability of where the electron is going to end up when we measure it. But talking about probabilities really only makes sense after the measurement. Jul 8, 2023 at 18:06
• @Omeglac You are exactly correct ... there is "information beforehand"! Excited electrons, that travel from the anode, or in the case of visible light that emit photons, are already interacting with the EM field .... long before emission. An electron can be exctted for us, ms, s time periods depending on the situation. Jul 9, 2023 at 12:40
• @Omeglac Feynman defined the path integral method to calculate virtual EM field strengths .... the math results in a similar way to the classical slit math. Probability paths are calculated and show the pattern. Jul 9, 2023 at 12:43
• @LenardKasselmann "collapse of the wave function .... still don't really understand"; if we make the classical assumption that the particle is created and nothing precedes the particle than yes you are correct ... it will always be a mystery and many scientists prefer that explanation! But if you consider the EM field more fully it is easy to understand the mystery ..... Feynman/Dirac said "the particle interferes with itself" but the modern interpretation is that the EM field is guiding the particle. Jul 9, 2023 at 12:50

if we fire electrons one by one through two slits we get an interference pattern

We get dots on the screen. These form an intensity distribution on the screen after the impact of many electrons.

the electron went through one of the slits "randomly" and we don't know which until we measure. But even if we don't measure it it still went through just one of those slits so why is there an interference pattern? What is interfering with the electron?

Good point. There is another player in the game. Somehow the obstacle of the slit should be included. Let's take a new view, where the surface electrons of the slit material interact with the passing electrons.
There are many observations of phononic inductions in materials, and this would be a good candidate to explain the swelling intensity on the screen.
The electron is not simply deflected by the slits, but redistributed according to the resonance with the phononic excitation.

Why doesn't looking at the screen count as an observation/measurement?

With the above explanation, yes.

• I would say that the obstacle of the slit causes the initial diffraction which is a necessary step before "interference" can occur. Interference is more likely the result if the wave nature of the dynamic EM field ..... even before the electron has left the electrode it is already interacting with the EM field ..... as are all the electrons in the apparatus. Jul 9, 2023 at 12:29

if the reality was that the electron really did go through both slits at the same time, how is this possible?

One possibility is that what we call an electron is actually a plasma-like collection of N+1 electrons and N positrons (where N can be a large number), and some of the particles go through one slit and some of them go through the other slit. Such a collection can have a charge density that approximates well ((in some sense) the standard charge density built from the wave function. Please see details of such model of quantum theory in my article Quantum Rep. 2022, 4(4), 486-508 and references there.

• This is not standard physics. I don't say this is wrong, but without mentioning the other, more prevailing interpretations of QM this shows a skewed picture. Oct 16, 2023 at 8:12
• @AccidentalTaylorExpansion "This is not standard physics." Maybe not, but there is nothing in the rules (physics.stackexchange.com/help/on-topic) about "non-standard physics", the rules ban "non-mainstream physics". The idea of my answer was published in three peer-reviewed papers, so I don't think it is "non-mainstream". And the answer begins with the words "One possibility is that...", so I told the readers about the limited scope of my answer. You are not happy about the "skewed picture", but there is nothing about "skewed" in the rules. This is just an answer, not a treatise. Oct 17, 2023 at 3:58