Given a state $|\psi \rangle$ one can form the expectation value of an observable $O$ as: $$ \langle \psi|O|\psi \rangle. $$ For the case $O = H$, where $H$ is the Hamiltonian of the quantum system, the expectation value above gives the expected energy of the state. Similarly, the quantum evolution of a state can be written as a map: $$ |\psi\rangle \to \mathrm{e}^{-iHt} |\psi \rangle = |\tilde\psi \rangle. $$ The expectation value $$ \langle \psi |\mathrm{e}^{-iHt} |\psi \rangle $$ thus gives the transition probability of $|\psi \rangle$ to $|\tilde\psi \rangle$. My question is: what is the interpretation of: $$ \langle \psi |O\mathrm{e}^{-iHt} |\psi \rangle? $$