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If the time reverse operator is defined as \begin{equation} \mbox{T}|\psi(t)\rangle=|\psi(-t)\rangle \end{equation} I am now considering time reversed $\hat x$ and $\hat p$ (of course in Heisenberg representation) \begin{equation} \hat x^T=T^\dagger U^\dagger\hat x\, U T=? \end{equation} \begin{equation} \hat p^T=T^\dagger U^\dagger\hat p\, U T=-? \end{equation} How can I evaluate RHS? Shankar is just stating that $\hat p^T=-\hat p$ and $\hat x^T=\hat x$. Is there a proof for it?

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I believe Shankar actually defines the quantum time reversal operator to be complex conjugation in the coordinate basis like as follows:

〈x|T|ψ〉=ψ*(x)

Check

〈x|TXT|ψ〉=〈x|X|ψ〉

and

〈x|TPT|ψ〉=〈x|-P|ψ〉

Note that T is self-adjoint, so TXT and TPT express transformations of X and P into their time reversed forms.

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