The Meaning of Time in Quantum Mechanics

Abstract

We propose a research program to investigate the role of time in non-relativistic quantum mechanics. Although our proposal is of general relevance, we will specialize our applications to the field of quantum chemistry. We will begin by discussing the meaning of a “chemical event” and the importance of constructing time operators to understand the nature of chemical events. We formulate a novel set of non-linear eigenvalue equations, which can be solved numerically to construct time operators for different many-body Hamiltonians. Next, we discuss quantum clocks and the conditions an ideal clock must satisfy. We introduce the adiabatic time approximation (ATA), which provides a rigorous criterion for time to be regarded as a parameter rather than an operator. Finally, we discuss the effects of decoherence on quantum clocks, within the framework of the theory of open quantum systems.

Project Summary

Since the early days of quantum theory, time has played a special role. While the spatial location of quantum objects can be easily manipulated, time appears to continuously and uncontrollably move forward. Therefore, the special role of time in quantum mechanics is often taken for granted, particularly in the field of quantum chemistry.

In our proposed research, we reexamine the meaning of time in quantum mechanics and question whether it truly deserves a special role. We look at the conditions under which time must be treated as a dynamical variable like space, and when it can in fact be treated as a special parameter. This naturally leads us to investigate the nature of “quantum clocks,” which are devices used to keep track of time. This research is significant, because it will raise and answer new questions about the meaning of time, which are not often addressed in current research.

Proposed Research

Introduction

The wave function of a quantum system describes the probability amplitude for different events to occur at points in space-time. However, the traditional formulation of quantum theory treats space and time on a completely different footing. In non-relativistic quantum mechanics, and chemistry in particular, it is often taken for granted that time is a continuous parameter and not a dynamical variable. If we think of space on the other hand, we know that position is treated as a dynamical variable represented by an operator.

The unequal treatment of space and time in quantum mechanics has many ramifications, particularly when we begin to ask questions about when and 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 the microscopic realms of quantum chemistry and condensed matter physics has hardly been explored to date.

I propose a FQXI-supported 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 three broad areas of research, outlined in the sections below.

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 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.

\label{full_t}

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}

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.

\label{full_H}

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 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}

where the first three terms represent free electrons, nuclei and photons, while the last three are their respective interactions. We can construct a time operator 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, as well as the domain expertise available to accomplish this task. 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?

\label{h}

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}. 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}}{2m}\nabla^{2}+v_{ext}(\mathbf{r},t)\right]\Psi(\mathbf{r})\\ +\int d^{3}r\int d^{3}r^{\prime}\Psi^{+}(\mathbf{r}^{\prime})\Psi^{+}(\mathbf{r})\left[\frac{e^{2}}{|\mathbf{r}-\mathbf{r}^{\prime}|}\right]\Psi(\mathbf{r})\Psi(\mathbf{r}^{\prime}),\\ \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 Heisenb