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In quantum mechanics, when we want to introduce reference frame change, we do it such that $\left|\psi'\right> = U\left|\psi\right>$. Using the fact that $\left<\psi|\psi\right>=1$, we then deduce that $U$ should be unitary.

My question is, how do we know that the transformation is a linear operator? Is this a postulate of quantum physics? Is it completely general?

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Short answer since I'm on mobile:

No, it's not a postulate but rather a theorem. First, clear things up a little. We want a symmetry of the theory to act as an arbitrary transformation which conserves the unitarity of our theory. A transformation which does not act as a symmetry of our system need not to be a linear transformation. Now, for symmetries, there exists a famous theorem by the mathematican Eugene Wigner, known as Wigner's Theorem, which states that the transformation encoding the action of this symmetry on the Hilbert space of states must be a linear transformation.

So to use your example, merely a change of reference frames is of course a symmetry and therefore by Wigner's theorem acts simply as a linear transformation.

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