[tfgg]

Could you elaborate why you think it looks like a fractional FT?

If you solve Schrödinger's equation with periodic boundary conditions on the potential what you get are Bloch functions, which are the product of a phasor and a periodic function.

[fallingfrog]

I thought so because the evolution of the waveform looked exactly like this: https://upload.wikimedia.org/wikipedia/commons/e/e3/FracFT_R... (from the wikipedia page on a fractional fourier transform)