# Time reversal in quantum mechanics

If the time reverse operator is defined as $$\mbox{T}|\psi(t)\rangle=|\psi(-t)\rangle$$ I am now considering time reversed $\hat x$ and $\hat p$ (of course in Heisenberg representation) $$\hat x^T=T^\dagger U^\dagger\hat x\, U T=?$$ $$\hat p^T=T^\dagger U^\dagger\hat p\, U T=-?$$ 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?

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.