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The unequal treatment of space and time in quantum mechanics has many ramifications, particularly when we begin to ask questions about \textit{when} and \textit{where} a particular event occurs. The implications of this have been explored extensively in the quantum gravity literature and in fields related to the foundations of quantum mechanics. However, the unequal treatment of space and time in quantum chemistry and condensed matter physics has hardly been explored to date.   I propose a research program that will explore the meaning of time in chemical systems. My group will explore questions such as: What is a chemical event? How can we define a clock to measure when a chemical event occurs? What are the conditions under which it is valid to treat time as a parameter and not a dynamical variable? Can the role of time in chemistry teach us about the meaning of time in other fields such as quantum gravity? To better understand these questions, we will explore four broad areas of research, outlined in the sections below.  %A detailed description of the proposed research, not to exceed 15 single-spaced 11-point pages, including a short statement of how the application fits into the applicant's present research program, and a description of how the results might be communicated to the wider scientific community and general public  %For instance, the position of an event and the momentum exchanged during an event cannot both be measured with infinite precision, as a consequence of canonical commutation relations between the position and momentum operators. Although time and energy satisfy an uncertainty principle as well, time is treated as a parameter and therefore no canonical commutation relations with the Hamiltonian exist. Another implication of the unequal  
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\subsection{Time Operators in Chemistry and the Meaning of "Chemical Events"}  Time-dependent processes in chemistry are ubiquitous. Chemical reactions occur as molecules collide with one another, exchanging kinetic and potential energy as chemical bonds are formed and broken. Similarly, in photochemical processes photons are absorbed and emitted as energy is channeled into different degrees of freedom (electronic, rotational and vibrational). For instance, the formation of a chemical bond or photodissociation of a molecule can be regarded as a "chemical event" and it is natural to ask not only where this event occurs, but also \textit{when}. In order to address this question, time must be regarded as an observable quantity, i.e. a dynamical variable represented by an operator and not a parameter.  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}  \left[\hat{t}, \hat{H} \right] |\psi_n \rangle= i \hbar |\psi_n \rangle,   \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},  \end{equation}  which describes a particle of charge q in one-dimension subjected to a constant electric field E. By solving eq~\ref{full_t}, one finds that the time operator is:  \begin{equation}  \hat{t} = -\frac{\hat{p}}{qE},  \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.  Solving eq.~\ref{full_t} for many-body systems is far less trivial. We must take into account correlations between different particles as well as fermonic or bosonic statistics. We propose to extend the above analysis to \textbf{construct time operators for chemical systems and study the chemical events they describe}. As a starting point, 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}  where the first three terms represent free electrons, nuclei and photons, while the last three are their respective interactions. A time operator is then constructed by solving the non-linear eigenvalue problem posed in eq.~\ref{full_t}. Unlike the single particle case, which can be solved exactly, solving the non-linear eigenvalue problem in eq.~\ref{full_t} must be done numerically for the Hamiltonian in eq.~\ref{full_H}. Non-linear eigenvalue problems are common in computational chemistry and many highly efficient numerical algorithms exist. My group has vast computational resources available to accomplish this task. \textbf{We will numerically construct time operators for a variety of chemical systems and investigate their spectra to answer questions about chemical events such as: When does a molecule absorb or emit a phonton? When does the absorbed energy fully dissipate into nuclear vibrational energy? How long does it take for a chemical reaction between two molecules to occur?}  The task of constructing a time operator for the full Hamiltonian in eq.~\ref{full_H} will be computationally demanding. Approximate and model Hamiltonians are often used to simply quantum many-body calculations. Each model Hamiltonian gives rise to a corresponding model time operator through the solution to eq.~\ref{full_t}. \textbf{Another area of research will be to construct and classify time operators for different model Hamiltonians of varying degrees of complexity.} For example, in chemistry it is often possible to make approximations to simplify eq.~\ref{full_H}. By invoking the Born-Oppenheimer approximation (BO), one can treat the nuclei classically as well as the electromagnetic field and arrive at the electronic Hamiltonian:  \begin{equation}  \hat{H} = \int d^3 r \Psi^+(\mathbf{r})\left[ -\frac{\hbar^2}{2 m} \nabla^2 + v_{ext}(\mathbf{r},t) \right] \Psi(\mathbf{r}) + \int d^3 r \int d^3 r' \Psi^+(\mathbf{r}') \Psi^+(\mathbf{r})\left[ \frac{e^2}{|\mathbf{r} - \mathbf{r}'|} \right] \Psi(\mathbf{r}) \Psi(\mathbf{r}'),  \label{h}  \end{equation}  where $\Psi^+(\mathbf{r})$ and $\Psi(\mathbf{r})$ are field operators, respectively creating and destroying an electron at position $\mathbf{r}$. The "external potential," $v_{ext}(\mathbf{r},t)$, is the potential external to the electrons, which incorporates the nuclear coulomb potential and the interaction between electrons and the electromagnetic field. The Hamiltonian in eq.~\ref{h} is often used as the starting point in chemistry and many-body physics. Therefore much of our research will focus on constructing time operators for this Hamiltonian with different external potentials, $v_{ext}(\mathbf{r},t)$.   In addition, it is often insightful to make further approximations and construct model Hamiltonians with discrete Hilbert spaces. One widely used example is the Hubbard model Hamiltonian:  \begin{equation}  \hat{H} = \sum_{i,j, \sigma} \left[ T_{ij} + v_{ext, i,j} \right ]\hat{c}_{i,\sigma}^+ \hat{c}_{j,\sigma} + \sum_{i} U_i \hat{c}_{i, \uparrow}^+ \hat{c}_{j,\uparrow}\hat{c}_{i, \downarrow}^+ \hat{c}_{j,\downarrow},  \end{equation}  where $\hat{c}_{i,\sigma}^+$ and $\hat{c}_{i,\sigma}$ respectively create and destroy and electron with spin $\sigma = \uparrow,\downarrow$ on the ith site. Another example is the Heisenberg Hamiltonian:  \begin{equation}  \hat{H} = \sum_{i,j} J_{ij} \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_j + \sum_{i} \mathbf{B}_i(t) \cdot \hat{\mathbf{S}}_i,  \end{equation}  which exclusively treats the spin degrees of freedom $\hat{\mathbf{S}}_i$, on the ith site in the presence of a local magnetic field $\mathbf{B}_i(t)$. Studying and classifying the time operators for these different Hamiltonians will be an important component of our research program.  %satisfying canonical commutation relations with the chemical Hamiltonian and not a parameter. I propose to e will construct time operators  Time-dependent processes in chemistry are ubiquitous. Chemical reactions occur as molecules collide with one another, exchanging kinetic and potential energy as chemical bonds are formed and broken. Similarly, in photochemical processes photons are absorbed and emitted as energy is channeled into different degrees of freedom (electronic, rotational vibrational). For instance, the formation of a chemical bond or photodissociation of a molecule can be regarded as a "chemical event" and it is natural to ask not only where this event occurs, but also \textit{when}. In order to address this question, time must be regarded as an observable quantity, i.e. a dynamical variable represented by an operator and not a parameter.  Mathematically, the problem of constructing the 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: