Group Theory why transformations of Hamiltonian are unitary?

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The discussion centers on proving that the transformation matrix D, which modifies the wavefunction, must be unitary to preserve certain properties. It is noted that transformations do not inherently need to be unitary unless they involve preserving vector lengths. The transformation is represented as Tψ(r) = ψ(Ur) = ΣDij ψ(r), indicating a relationship between the wavefunction and the matrix D. The participants are exploring whether the requirement for unitarity applies only when scaling the position vector r. The conversation emphasizes the need for a proof to establish the unitarity of D in this context.
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This is what I have so far:

part1.png
part2.png


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

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