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Master Equations

Robert W. Zwanzig

Published Apr 19, 2001• 1 citation
Publisher

Abstract

Abstract The Golden Rule is a colloquial name for a standard formula in time-dependent quantum mechanics. It provides a way to calculate the rate of transition between quantum states. It is approximate and has limited applicability; however, it is the starting point for many treatments of rate processes. Later it will be used in the discussion of master equations. This section gives the usual derivation of the Golden Rule. The following section shows how it is used to relate optical absorption spectra to time correlation functions.

DOI
10.1093/oso/9780195140187.003.0003

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Abstract The Golden Rule is a colloquial name for a standard formula in time-dependent quantum mechanics. It provides a way to calculate the rate of transition between quantum states. It is approximate and has limited applicability; however, it is the starting point for many treatments of rate processes. Later it will be used in the discussion of master equations. This section gives the usual derivation of the Golden Rule. The following section shows how it is used to relate optical absorption spectra to time correlation functions.
Abstract Fermi's Golden Rule is one of the most important results in quantum mechanics, predicting transition rates in applications ranging from beta decay to optical transitions, autoionization and photoionization. Most textbooks derive the Golden Rule only for the canonical case of a discrete state coupled by a time-independent perturbation to a continuum of final states. Here we present a unified treatment of several physical situations, including light absorption in atoms and solids, that all lead to the same Golden Rule expression. The essential ingredients are the transition matrix element and a spectral weighting function, whose physical origin need not be the density of final states, nor necessarily anything to do with the matter.The conventional discrete-to-continuum derivation introduces pedagogical complications associated with continuum normalization and the interpretation of the density of states. Among the cases considered here, the derivation for discrete transitions driven by a noisy perturbation is no more difficult mathematically but conceptually clearer, while leading directly to practical calculations. This suggests that it provides a more natural first introduction to the Golden Rule, with conventional continuum derivations better viewed later as another realization of the same underlying mathematical structure, rather than the other way around.
A straightforward extension of time-dependent quantum-mechanical perturbation theory is made to obtain the equivalent formulae for a statistical mixture of states. The 'master equation' resulting from this procedure is, in its lowest order, the analogue in quantum statistics of the 'Golden Rule' for transition rates and reduces exactly to this form in the limit of pure initial states. The authors give results to fourth order and illustrate their use by applying them to the problem of the resonant interaction of an electromagnetic field and a two-level atom.
Abstract Chapter 14 covers the statistical mechanics of driven quantum systems and the non-equilibrium ensembles associated with them. The topic is presented both from the state-vector and density matrix perspectives. The chapter begins with a discussion of driving forces resulting from light-matter interactions and their treatment via time-dependent perturbation theory. The Fermi Golden Rule is derived, and it is shown how to incorporate the Golden Rule into the calculation of frequency spectra within the canonical ensemble, which then connects to canonical quantum time correlation functions. The formalism is derived again from the quantum Liouville equation. The discussion then turns to the challenge of computing quantum time correlation functions and approximate techniques based on the Feynman path integral of Chapter 12. These schemes include centroid and ring-polymer molecular dynamics as well as open-path integral methods
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
Abstract Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from − ∞ to + ∞ , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time , even when the first order perturbation approximation itself is still valid.
Abstract Putting a general, physically relevant upper bound on equilibration times in closed quantum systems is a recently much pursued endeavor. In 2017 Phys. Rev. X 7 031027 García-Pintos et al suggest such a bound. We point out that the general assumptions which allow for an actual estimation of this bound are violated in cases in which Fermi’s golden rule and related open quantum system theories apply. To probe the range of applicability of Fermi’s golden rule for systems of the type addressed in the above work, we numerically solve the corresponding Schrödinger equation for some finite spin systems comprising up to 25 spins. These calculations shed light on the breakdown of standard quantum master equations in the ‘superweak’ coupling limit, which occurs for finite sized baths.