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I expand my comment into an answer. The idea is to fix $\alpha, \beta \in \mathbb R$ in order that, if $P:= U(\alpha, \beta)$ (which is automatically unitary), we have (i) $PP=e^{ik}I$ for some $k\in \mathbb R$, (ii) $P\hat{x}P^\dagger = -\hat{x}$, (iii) $P\hat{p}P^\dagger = -\hat{p}$ Since $\hat{x}$ and $\hat{p}$ has to be treated symmetrically, we ...



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