Some books argue that typical coordinate transformations such as space translations and rotations are represented in quantum mechanics by unitary operators because the Wigner's theorem. However I do not find any clear proof of this. For instance, suppose 1D for the sake of simplicity, by definition spatial translations change the position operator as X→X+a where "a" is a constant. I would want to prove that this transformation between operators can be implemented by a unitary operator. To this end, I try to apply the Wigner's theorem showing that the change X→X+a on the position operator induce a change in the state vectors |ψ>→|ψ'> which is one-to-one and preserves transition amplitudes, i.e. |<ψ|φ>|=|<ψ'|φ'>|. Any idea?