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(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A Why do space translations satisfy the Wigner's Theorem?

Tags:

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**