Main Question or Discussion Point
From many sources (Internet, Landau & Lifshitz, etc.), it is claimed that the Schrödinger's equation is a wave equation. However I do not understand why for the following reasons:
- It is Galilean invariant, unlike the wave equation which is Lorentz invariant. Note that the diffusion/heat equation is also Galilean invariant.
- If one takes the free particle localized in a finite region at time t0, then at any instant afterwards, the wavefunction will have non zero values arbitrarily far away from that region. I.e. there is a diffusion without any speed limit, of the wavepacket. That's another point making the Schrödinger equation looking more like the heat equation than the wave equation.
- Mathematically its determinant is such that the Schrödinger's equation qualifies as a parabolic PDE, same as the heat equation and unlike the wave equation (hyperbolic).