Many things can “happen” inside a quantum wave function, of course, including everything that actually does happen — formation of galaxies, origin of life, Lady Gaga concerts, you name it. But given a certain quantum wave function, what actual is happening inside it?
A surprisingly hard problem! Basically because, unlike in classical mechanics, in quantum mechanics the wave function describes superpositions of different possible measurement outcomes. And you can easily cook up situations where a single wave function can be written in many different ways as superpositions of different things. Indeed, it’s inevitable; a humble quantum spin can be written as a superposition of “spinning clockwise” or “spinning counterclockwise” with respect to the z-axis, but it can equally well be written as a superposition of similar behavior with respect to the z-axis, or indeed any axis at all. Which one is “really happening”?
Answer: none of them is “really happening” as opposed to any of the others. The possible measurement outcomes (in this case, spinning clockwise or counterclockwise with respect to some chosen axis) only become “real” when you actually measure the thing. Put more objectively: when the quantum system interacts with a large number of degrees of freedom, becomes entangled with them, and decoherence occurs. But the perfectly general and rigorous picture of all that process is still not completely developed.
So to get some intuition, let’s start with the simplest possible version of the problem: what happens inside a wave function (describing “system” but also “measurement device” and really, the whole universe) that is completely stationary? I.e., what dynamically processes are occurring while the wave function isn’t changing at all?
You’re first guess here — nothing at all “happens” inside a wave function that doesn’t evolve with time — is completely correct. That’s what I explain in the video above, of a talk I gave at the Philosophy of Cosmology workshop in Tenerife. The talk is based on my recent paper with Kim Boddy and Jason Pollack.
Surprisingly, this claim — “nothing is happening if the quantum state isn’t changing with time” — manages to be controversial! People have this idea that a time-independent quantum state has a rich inner life, with civilizations rising and falling within even though the state is literally exactly the same at every moment in time. I’m not precisely sure why. It would be more understandable if that belief got you something good, like an answer to some pressing cosmological problem. But it’s the opposite — believing that all sorts of things are happening inside a time-independent state creates cosmological problems, in particular the Boltzmann Brain problem, where conscious observers keep popping into existence in empty space. So we’re in the funny situation where believing the correct thing — that nothing is happening when the quantum state isn’t changing — solves a problem, and yet some people prefer to believe the incorrect thing, even though that creates problems for them.
Quantum mechanics is a funny thing.