Plane Waves in TDSE: Group & Phase Velocity

[sachi]

We are asked to substitute a plane wave solution into the TDSE for V=0 and show that it satisfies the TDSE (this is straightforward). Does this mean the plane waves are only solutions to the TDSE if V=0, or do they have other significance?

Also we show that the group velocity = v (the particle velocity) and that the phase velocity =v/2. The calculations are straighforward but the interpretation is difficult. I can see why the group velocity (the effective speed of the wave packet and the speed at which information is propogated) = v, but I can't see why the phase velocity = v/2.
Thanks

 

[Physics Monkey]

Yes, in general plane waves are only solutions of the Schrödinger equation in regions where the potential is constant. The further imposition of boundary conditions when the potential is a series of steps generally restricts these states further (unlike in the free space case where the potential is the same constant everywhere). You can understand this quite physically as follows. The plane waves are states of definite momenta, but you know that in the presence of a non-constant potential, the momentum will change, therefore these states can't have definite energy (they can't be energy eigenstates) when the potential varies with position.

The point about the phase velocity versus group velocity bit is simply that the group velocity is what corresponds most directly with what you are familiar with from classical mechanics.