Group Theory why transformations of Hamiltonian are unitary?

[applestrudle]

This is what I have so far:

I'm trying to show that the matrix D has to be unitary. It is the matrix that transforms the wavefunction.

 

[HallsofIvy]

The matrix that transforms the wave function how? So that it preserves some property? Transformations do NOT have to be unitary unless you are trying to lengths of vectors.

 

[applestrudle]

The matrix that transforms the wave function how? So that it preserves some property? Transformations do NOT have to be unitary unless you are trying to lengths of vectors.

In lectures we were showing
Tψ(r) = ψ(Ur) = ΣDij ψ(r)

Dij has to be unitary and form a representation of T - I'm just trying to figure out the proof. Are you saying this is only try if you scale the position vector r?