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International Journal of Quantum Chemistry

Time-dependent quantum statistical inequalities

Sidney Golden

Published Jun 18, 2009• 1 citation
Publisher

Abstract

The following is established. Any system having a temporal evolution which is determined by its Hamiltonian and which has an initial statistical operator that is a completely monotonic function of a given nonnegative and time-independent observable will evolve in time so that the expectation value of that observable is never less than the initial value. In a suitable time-averaged sense, for systems having time-independent Hamiltonians, the velocity and acceleration of the expectation value are shown to be nonnegative. The results obtained depend in no way on the system being macroscopic. A theorem dealing with the energy of a system and another dealing with the virial of a system are derived to illustrate the use of the time-dependent inequalities.

DOI
10.1002/qua.560070766

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The following is established. Any system having a temporal evolution which is determined by its Hamiltonian and which has an initial statistical operator that is a completely monotonic function of a given nonnegative and time-independent observable will evolve in time so that the expectation value of that observable is never less than the initial value. In a suitable time-averaged sense, for systems having time-independent Hamiltonians, the velocity and acceleration of the expectation value are shown to be nonnegative. The results obtained depend in no way on the system being macroscopic. A theorem dealing with the energy of a system and another dealing with the virial of a system are derived to illustrate the use of the time-dependent inequalities.
The time-dependent Schrödinger equation forms the chapter's starting point. For a time-independent Hamiltonian, a quantum state may be time evolved with the standard, unitary time evolution operator. After obtaining this operator, an algorithm is provided by which a state may be time evolved, and measurement probabilities determined. A spin 1/2 example illustrates time evolution concretely. All this makes close contact with the discussion of unitary operators in Chapter 10. The time dependence of expectation values forms the next topic, followed by the concept of a constant of the motion, which is carefully contrasted with that of a stationary state. The energy-time uncertainty relations, which rest on a quite different conceptual footing than the ‘standard’ uncertainty relations, occupy the rest of the chapter. An example from spin 1/2 illustrates the energy-time uncertainty relations.
We treat a quantum mechanical system with certain general properties which are expected to be common in macroscopic quantum systems. Starting from a PURE initial state (which may not describe an equilibrium) in which energy is mildly concentrated at a single value, we consider a time evolution determined by a time-dependent Hamiltonian as a model of an adiabatic operation in thermodynamics. We take a family of operations with the same procedure and various ``waiting times.'' Then the Minimum Work Principle is rigorously proved for almost all choices of the waiting time.
von Neumann has proved a quantum-mechanical ergodic theorem which deals with a closed macroscopic system and macroscopic observables. Certain unsatisfactory features of this theorem are probably due to the overly-strong idealization of a completely closed system.Since the claims of statistical mechanics go beyond macroscopic systems, there is a need for a more general theorem. The present paper deals with a system of any number of degrees of freedom and with any observable, during interaction with a temperature bath. In order to avoid an assumption of randomness, the bath must be described explicitly and precisely by a time-independent Hamiltonian. The model of the bath is obtained by taking the Gibbs ensemble seriously, i.e., as a set of NNN identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials VVV.
Recently, the notion of a quantum acceleration limit has been proposed for any unitary time evolution of quantum systems governed by arbitrary nonstationary Hamiltonians. This limit articulates that the rate of change over time of the standard deviation of the Hamiltonian operator — representing the acceleration of quantum evolution within projective Hilbert space — is constrained by the standard deviation of the time-derivative of the Hamiltonian, expressed as σ ̇ H ≤ σ H ̇ . In this paper, we extend our earlier findings to encompass any observable A within the framework of unitary quantum dynamics, leading to the inequality σ ̇ A ≤ σ v A . This relationship signifies that the speed of the standard deviation of any observable is limited by the standard deviation of its associated velocity-like observable v A . Finally, for pedagogical purposes, we illustrate the relevance of our inequality by providing clear examples. We choose suitable observables related to the unitary dynamics of two-level quantum systems, as well as a harmonic oscillator within a finite-dimensional Fock space.
We derive a generalization of the Ehrenfest theorem valid for open quantum systems. From this result, we identify three contributions to the evolution of expected values: (i) the explicit time dependence of the observable, (ii) the incompatibility between the observable and an operator which plays the role of an effective Hamiltonian, and (iii) entropy changes. Considering the local Hamiltonian as the observable, and adopting a specific interpretation of the nature of thermal interactions, we obtain an alternative version of the first law of thermodynamics. Within this framework, we show that in some cases the power performed by the system can be considered as a quantum observable. As an application, the pure-dephasing process is reinterpreted from this perspective.
Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and Rényi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in cases where the system density matrix obeys the Gorini-Kossakowski-Lindblad-Sudarshan equation.