The wavefunction is a tool to calculate the probabilistic results of experiments. It works well.
A lot of mystery surrounds what Ψ(x) actually is. Most confusion is self-inflicted. I urge my students to start with the pragmatic approach. Concentrate on what we know about the wavefunction and what we need to know. Then learn the theoretical and experimental facts. At that point you can sensibly consider adding an extra interpretation.
Other observations motivate it being a probability function:
- It is normalised to 1. If Ψ is a wavefunction. It satisfies Schrodinger’s equation but so does Ψ/2 because Schrodinger is a linear equation. But Ψ/2 is not a valid wavefunction. This normalisation of amplitude, rather than frequency, is not a property of classical wave solutions.
- It does not exist in space. Although it may look like a function of space for a single particle, for two particles it is an inseparable function of six spatial coordinates. technically, it is a function of configuration space however many dimensions that is.
- It changes instantly and non-locally when our knowledge changes. For a beam splitter its magnitude squared is 1/2 in each arm of the experiment, representing a 50:50 outcome. But once a particle is detected on the lefthand side the value on the right is immediately zero. There are never coincidences in both sides. Just as with a classical experiment with a two way divider hidden in a box.