We can’t see wave functions. It is a tool used to predict observations but itself cannot be observed, and cannot be an observable object as it exists in an abstract Hilbert space and not even in spacetime. It is only “space” in the sense of a state space, kind of like how if I have a radio with 4 knobs, I can describe the settings with a single point in a 4 dimensional space. That doesn’t mean the radio actually is a 4 dimensional object existing in this state space, it only means that we can represent that way for convenience, and the dimensions here moreso represent degrees of freedom.

If you believe everything is a wave function then you believe the whole universe is made out of things that cannot be observed. So how does that explain what we observe? Just leads to confusion. Confusion not caused by the mathematics but self-imposed. Nothing about the mathematics says you literally have to think everything is made out of waves. In fact, Heisenberg’s original formulation of quantum mechanics made all the same predictions yet this was before the Schrodinger equation was even invented.

People take the wave formulation way too literally and ultimately it just produces much of this confusion. They are misleadingly taught that you can think of things turning into waves by starting with the double-slit experiment, except it is horribly misleading because they think the interference pattern they’re seeing is the wave function. Yet, (1) the wave function is associated with individual particles, not the interference pattern which is formed by thousands, millions of particles. There is nothing wave-like visible with just a single particle experiment. (2) Even the interference pattern formed by millions of particles does not contain the information of the wave function, only a projection of it, sort of like its “shadow” as the imaginary terms are lost when you apply the Born rule to it and square it. (3) They also like to depict a literal wave moving through two slits, but again there are imaginary components which don’t map to anything physically real, and so the depiction is a lie as information has to be removed in order to actually display a wave on the screen.

The moment you look at literally anything that isn’t the double-slit experiment, the intuitive notion of imagining waves moving through space breaks down. Consider a quantum computer where the qubits are electrons with up or down spin representing 0 or 1. You can also represent the state of the quantum computer with a wave function, yet what does it even mean to imagine the computer’s internal state is a wave when there is nothing moving at all and the state of the quantum computer doesn’t even have position as one of its values? You can’t point to that wave even existing anywhere, you get lost in confusion if you try.

This cloud is described by a mathematical object called wave function. The Austrian physicist Erwin Schrödinger has written an equation describing its evolution in time. Quantum mechanics is often mistakenly identified with this equation. Schrödinger had hopes that the ‘wave’ could be used to explain the oddities of quantum theory: from those of the sea to electromagnetic ones, waves are something we understand well. Even today, some physicists try to understand quantum mechanics by thinking that reality is the Schrödinger wave. But Heisenberg and Dirac understood at once that this would not do.

To view Schrödinger’s wave as something real is to give it too much weight – it doesn’t help us to understand the theory; on the contrary, it leads to greater confusion. Except for special cases, the Schrödinger wave is not in physical space, and this divests it of all its intuitive character. But the main reason why Schrödinger’s wave is a bad image of reality is the fact that, when a particle collides with something else, it is always at a point: it is never spread out in space like a wave. If we conceive an electron as a wave, we get in trouble explaining how this wave instantly concentrates to a point at each collision. Schrödinger’s wave is not a useful representation of reality: it is an aid to calculation which permits us to predict with some degree of precision where the electron will reappear. The reality of the electron is not a wave: it is how it manifests itself in interactions

— Carlo Rovelli, “Reality is Not What it Seems”

It is more intuitive to not think of wave functions as entities at all. But people have this very specific mathematical notation so burned into their heads from the repeated uses of the double-slit experiment that it is very difficult to get it out of their heads. Not only did Heisenberg instead use matrix transformation rather than the Schrodinger equation to represent QM, but it is also possible to represent quantum mechanics in even a third mathematical formulation known as the ensemble in phase space formulation.

The point here is that the Schrodinger equation is just one mathematical formalism in which there are multiple mathematically equivalent ways to formulate quantum mechanics, and so treating these wave functions wave really existing waves moving through a Hilbert space which you try to imagine as something like our own spacetime seems to be putting too much weight on a very specific formalism and ultimately is the source of a lot of the confusion. Describing the whole universe as thus a giant wave in Hilbert space evolving according to the Schrodinger equation is thus rather dubious, especially since these are entirely metaphysical constructs without any observable properties.