Unitary time-evolution operator on position and momentum eigenstate

It depends on the Hamiltonian of a system as $$U(t)=e^{-iHt/hbar}$$

To know $$U(t)|xrangle mathrm{or} U(t)|prangle$$

One must know $$H|xrangle mathrm{or} H|prangle $$

For example, suppose a free particle $$H=frac{P^2}{2m}$$ $$U(t)|prangle=e^{-iHt/hbar}|prangle=e^{-ip^2/2mhbar}|prangle$$

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Edit: The last line in question $$U(t)|xrangle=f(t)|xrangle$$ so that matrix element of $U(t)$ in position basis $$U(x,t;x')=langle x'|U(t)|xrangle = f(t)langle x'|xrangle=f(t)delta(x-x')$$ which is indeed wrong! (Think why yourself)