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Is the preservation of the inner product the same thing as the vector length of the wave-function staying constant with it's rotation through some R2 plane (ie it's evolution through time), that is when trying to explain the reasons why a normalized wave-function stay normalized over the evolution of time?

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  • $\begingroup$ I'm not sure, exactly, what it is that you want to know. Anyway, the inner product remains conserved during time evolution because of probability conservation/unitarity. See my answer here: physics.stackexchange.com/a/434912/133418 $\endgroup$ – Avantgarde Jul 3 at 20:00
  • $\begingroup$ @Avantgarde what I meant is that can we think of the inner products staying conserved as the magnitude of the wave-function vector staying conserved with its rotation within a vector space? $\endgroup$ – EPIC Tube HD Jul 4 at 4:38
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Not exactly, but something similar. Quantum states preserve normalization through unitary transformations.

These are the transformations that represent a symmetry of the system. So, in some sense, unitary transformations represent a change in reference frame. Since physics should not change according to reference frame, the probabilities should be the same in all frames. This, in turn, implies conservation of normlization.

Rotation is one example of change of reference frame, which means there is a corresponding unitary transformation for that. But these transformations are not limited to rotations. Translations (both in time and space) are also symmetries of the system (because there is no prefered point in spacetime for the origin), so there are unitary transformations for that too.

(Notice that to apply a time translation in hilbert space we do what you call 'time evolution')

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