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> But that's no more valid than saying that a regular water wave is "just a function that assigns a height to the water at every (x, y, t) co-ordinate, it doesn't go anywhere".

I think this is a very important distinction actually. A wave amplitude represents an actual displacement in some medium, and waves interfere constructively/destructively because they both move the medium in the same/opposite direction at the same time at the same location. So when a water wave gets pushed through two slits, it breaks into two separate water waves, one coming from each slit, and those two waves push the water up and down at the same time at different locations.

But wavefunctions are very much not like that. A wavefunction amplitude does not represent a displacement in any kind of medium. They represent a measure of the probability that the system being described is in a particular state at a moment in time. That state need not even be a position, it might be a charge, or a spin, or a speed, or any combination of these. Basically quantum systems oscillate between their possible states, they don't oscillate in space-time like matter affected by a wave does.

This also makes it very hard to conceptualize what it means for these wavefunctions to interfere. So the simple picture of "wave A and wave B are pushing the water up at the same time in the same location, so the water rises higher when both waves are there" is much harder to apply to probability oscillations than a direct comparison makes it out to be.

An additional problem when comparing wavefunctions to waves in a medium is that there is no source of a wavefunction. Any system you're analyzing has a single wavefunction, that assigns probability amplitudes to every possible configuration of that system. You can decompose the system's wavefunction as a sum of multiple wavefunctions corresponding to certain measurables, but this is an arbitrary choice: any such decomposition is exactly as valid. In matter waves, if I drop two stones in water at different locations, the water surface's movements can be described as a single wave, but there is a natural decomposition into two interfering waves each caused by one of the stones. There is no similar natural decomposition that quantum mechanics would suggest for a similar quantum mechanical system.

> when a water wave gets pushed through two slits, it breaks into two separate water waves, one coming from each slit, and those two waves push the water up and down at the same time at different locations.

What physically observable distinction are you drawing? The points of water on the far sides of the slits will have a certain height at each point at each time, forming the interference pattern you'd expect. You can decompose that function into a sum of two separate waves, if you want, but you don't have to. And exactly the same thing is true of the quantum mechanical wavefunction for a particle passing through a pair of slits.

> A wavefunction amplitude does not represent a displacement in any kind of medium. They represent a measure of the probability that the system being described is in a particular state at a moment in time. That state need not even be a position, it might be a charge, or a spin, or a speed, or any combination of these. Basically quantum systems oscillate between their possible states, they don't oscillate in space-time like matter affected by a wave does.

I don't think that's a real distinction. Water height is a different dimension from (x,y) position and it behaves very differently; that the wave is moving across the surface and that the surface is moving up and down are orthogonal facts, the reason the former is movement isn't that the latter is movement. A classical electromagnetic wave moves even though it isn't in a medium that's moving (and so does e.g. a fir wave).

> You can decompose the system's wavefunction as a sum of multiple wavefunctions corresponding to certain measurables, but this is an arbitrary choice: any such decomposition is exactly as valid. In matter waves, if I drop two stones in water at different locations, the water surface's movements can be described as a single wave, but there is a natural decomposition into two interfering waves each caused by one of the stones. There is no similar natural decomposition that quantum mechanics would suggest for a similar quantum mechanical system.

Again I don't think this is a real distinction. You have exactly that natural decomposition in the QM system - it's not the only valid decomposition, but it is a valid one and it has some properties that make it nice to work with. And similarly for dropping stones in the water, infinitely many other decompositions are possible and equally valid (e.g. decomposing as two copies of a wave where you dropped two half-sized stones into the water).

In classical mechanical waves like water waves or sound waves, the wave equations are just an approximation of the "real" movement happening. You can theoretically get the precise results of the movement of every part of the medium by just applying Newton's laws of motion to all of the particles of the medium (assuming they're coarse grained enough that you can ignore quantum effects, of course). That is, you can model the surface of the water and the rocks as little balls of mass connected by ideal springs and such, and you'll see that the movement of the water is directly caused by the two rocks, whose contact with the water pushes it here and there. And so, the "real" picture is two separate waves, and any other way to model the system, even if mathematically equivalent, is artificial in some sense.

Even for EM waves, the classical theory explains them somewhat mechanistically, as an interaction between electrical and magnetic forces that originate from the charged sources and self-propagate.

There is no similar picture you can draw for the quantum mechanical wavefunction. It's the base reality of the system, and it turns out in fact that Newton's laws can be derived as an approximatiom of the wavefunction. But there isn't any kind of "reason" for which the wavefunction does what it does, like there is for the water waves. And so all models for separating the wavefunction into different components is just as "natural" as any other.

> you can model the surface of the water and the rocks as little balls of mass connected by ideal springs and such, and you'll see that the movement of the water is directly caused by the two rocks, whose contact with the water pushes it here and there. And so, the "real" picture is two separate waves

I don't see that that follows? Both wave pictures are a mathematical approximation to the "real" system of particles moving. Both are ways of calculating the same result. Neither is objectively more valid than the other. You can say that picturing it as "the wave from the first rock overlaid on the wave from the second rock" is an interpretation that makes more physical sense or is nicer to think about, but that's just as true in the QM case.

> Both wave pictures are a mathematical approximation to the "real" system of particles moving.

Yes, exactly.

> You can say that picturing it as "the wave from the first rock overlaid on the wave from the second rock" is an interpretation that makes more physical sense or is nicer to think about, but that's just as true in the QM case.

No, because in QM, the wavefunction is the real, final picture. Any split is arbitrary, the two rocks didn't "cause" the waves, the the wavefunction of all the molecules that make up the rocks and the water has higher amplitudes for states in which the water molecules look like the wavey surface, and lower amplitudes for states in which the rocks touch the water but the water doesn't move at all, etc. That's all the physics can say.

> No, because in QM, the wavefunction is the real, final picture. Any split is arbitrary, the two rocks didn't "cause" the waves

But at the wave picture level that's always true. You can break down a water wave into a bunch of particles moving according to forces, sure. But that doesn't actually help you answer the question of whether a wave that's gone through a pair of slits is now one wave or two waves, because when you're looking at the particles and forces you can't see the waves any more (except as patterns in the particles and forces - but then you're back in the wave picture).

I agree things like the slit experiments are easily explained by classical waves, and thus seem to point to particles being waves; and I see no inherent reason to bring up probability distributions and such to answer that observation.

Where I'm struggling is that classical waves will always spread out spherically, and their energy must do so too. The issue here being that if a photon is a minimal quanta of energy, but is just a classical wave, what prevents it from spreading out and having sub-photon energy? Or if indeed it does, how does that sub-photon quantity get measured? -- if these experiments claim to be emitting a time-series of single photons, classical wave interference won't occur (again, being separated in time).

I don't know how fringe this is, but Huygens Optics gives a possible answer here: https://www.youtube.com/watch?v=tMP5Pbx8I4s. TL;DW: if you assume certain non-linear properties of vacuum as a medium, it seems possible for light (EM waves) to self-contain spatially instead of spreading out, and such trapped energy seems, for all intents and purposes, to behave like particles.

Again, IANAPhysicist, so I don't know what to think of that video, but the channel seems legit, and the explanation is beautiful in its simplicity.

> The issue here being that if a photon is a minimal quanta of energy, but is just a classical wave, what prevents it from spreading out and having sub-photon energy? Or if indeed it does, how does that sub-photon quantity get measured? -- if these experiments claim to be emitting a time-series of single photons, classical wave interference won't occur (again, being separated in time).

Right, so that part is new and "spooky" - quantum phenomena are quantised (hence the name). The photon does spread out as a wave, there is in a sense half a photon heading towards the top half of the screen and half a photon heading towards the bottom half of the screen - but then when it hits the screen what we see is a single whole photon that hits either the top half or the bottom half (or, perhaps, half of an us sees a photon hit the top half and half of an us sees a photon hit the bottom half). This is the "wave-particle duality" and while it does fall out of the equations, it's definitely unfamiliar compared to classical physics.

If you want to fully understand, all I can suggest is "work your way through a QM textbook" - every popular explanation I've seen has messed it up one way or another. But it sounds like you're understanding correctly, and thinking for yourself as well - you've hit upon the actual essence of it, the kernel that really is hard.

My personal interpretation is that elementary particles physically are those waves and never anything else. Those waves interact with each other with probabilistic events exchanging some energy and momentum and reshaping each other. They can get narrowed down if they exchange energy/momentum or get spread apart, for example through interactions on the edge of an object. What's completely virtual for me is the idea of pointlike particles occupying some specific location and having some specific momentum. Almost everything we know contradicts this idea, and yet we cling to it.

I think Craig Bohren wrote in one of his books, that to get anything calculated and done the waves are all you need. Particles are nice for some kind of visualization, but they don't really help getting things done. I liked that.

Instead of particles I like to view the interactions like the forming of a lightning in a thunderstorm. The energy-field builds up, And at some point of contact the energy is being released in a single lightning strike.

What I still wonder is, if the interaction really depletes the energy-field instantly in a single point, or if there is more going on (on different timescales - maybe with speeds not related to the speed of light).