On pg 72 of "Something Deeply Hidden," Sean Carroll discusses how the uncertainty principle is just a consequence of the Schrodinger equation. He writes:

Consider a simple sine wave, oscillating up and down in a regular pattern throughout space. Plug such a wave function into the Schrodinger equation and ask how it will evolve. We find that a sine wave has a definite momentum, with shorter wavelengths corresponding to faster velocity. But a sine wave has no definite position; on the contrary, it's spread out everywhere.

I'm interested in how you describe. Carroll doesn't describe exactly what happens to a wave function that looks like a sine wave when you let it evolve under the Schrodinger equation. It seems like he might be saying it stays a sine wave, but I can't tell. What happens to a sine wave when you let it evolve under the Schrodinger equation?

If anyone knows of any kind of animation of this, that would be awesome.