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Mathematically, the problem of constructing time operators is formulated as follows: Given a Hamiltonian $\hat{H}$, its associated time operator $\hat{t}$, is constructed by imposing the canonical commutation relation:  \begin{equation}  \label{full_t}  \left[\hat{t}, \hat{H} \right] |\psi_n \rangle= i \hbar |\psi_n \rangle,   \end{equation} \end{equation}~\label{full_t}  over a certain domain of the Hilbert space spanned by the states $\{ |\psi_n \rangle \}$. Note that eq.~\ref{full_t} is a non-linear eigenvalue problem, as one must find both the operator $\hat{t}$ and the eigenvectors $\{ |\psi_n \rangle \}$. Solving eq.~\ref{full_t} for single particle quantum systems is often a fairly straight forward task. For example, consider the Hamiltonian:  \begin{equation}  \hat{H} = \frac{\hat{p}^2}{2m} - qE\hat{x},