I've read every thread on StackExchange (and Quora and reddit...) that I can find about a physical intuition for the phase in the quantum wave function, and I still Just. Don't. Get. It. (Yes, I've seen this thread--didn't help!)
As a jumping off point, I've been watching this terrific visualization of the quantum wave function. According to this video, for a particle in an infinite square well, the "phase" will rotate in the complex plane. Okay...what does that mean physically? Let's focus on just the ground state wave function. If it's not "rotating" in real space (right?), then what exactly is changing to make the phase "rotate"? If I could "see" the wave function with my eyes, what would I see?
I understand the mathematical argument that the phase doesn't matter: the complex exponential cancels out when you calculate the probability distribution, etc.
Maybe my confusion stems from a misunderstanding of what phase even is in quantum mechanics. When I visualize phase, I think of a sine wave and how much it has been shifted to the left or right (relative to some origin). But when I watch that visualization of the ground state wave function, nothing is sliding left or right, the wave isn't going anywhere. So what information does the phase encode here? I'm clearly missing something...
As background, I'm fairly new to quantum physics. I've always wanted to understand it beyond the usual "pop sci" descriptions, so I've been following the MIT OpenCourseware lectures on quantum physics. The physical nature of this phase is just really tripping me up and I've yet to find any explanation anywhere that goes beyond "well it works out in the math." Or is that all it is: just a convenient mathematical bookkeeping trick that physicists keep around because it happens to match observations?