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\end{equation}  and the states $\{ |\psi_n \rangle \}$ span the entire Hilbert space. The eigenstates of $\hat{t}$ describe states that arrive at the origin ($x=0$) at a definite time. Thus the "event" defined by the time operator for the linear potential is arrival at the origin.  A slightly less trivial example is the We propose to construct extend the above analysis to \textbf{construct  time operators for chemical systems and study the chemical events they describe.To describe}.To  construct time operators for chemical systems, one might begin with the full Hamiltonian operator of non-relativistic electrons and nuclei, interacting with the quantized electromagnetic field: \begin{equation}  \hat{H} = \hat{H}_{el} + \hat{H}_{n} + \hat{H}_{em} + \hat{H}_{el-n} + \hat{H}_{el-em} + \hat{H}_{n-em},  \end{equation}~\label{full_H}