Introduction to the density operator Motivation and definition It is often inconvenient, for one reason or another, to keep track of all the variables of a complex system. For example, in a many-body system it would be impracticable to consider the co-ordinates of each particle. Furthermore, such information is actually not of much interest. Thus in a spin system the behaviour of the individual magnetic moments is unimportant; the components of the total magnetisation are the variables of primary interest. Very generally, for a system with ∼ 10 23 co-ordinates, one is unlikely to need more than, say, ten variables to describe its observable properties. However, it is quite clear that this reduced amount of information is no longer sufficient to write down a wavefunction and, therefore, it is no longer possible to calculate the evolution of such a system using the usual methods of quantum mechanics. In Chapter 5 we saw that it was possible, by the introduction of probabilistic arguments, to calculate the evolution of the total magnetisation of a spin system. The calculations were performed using the machinery of quantum mechanics and meaningful and useful results were thereby obtained. In this chapter we will look at things from a rather different point of view. We will not, initially, be concerned with the expectation values of certain specified observables, rather we direct our attention to the general description of the ‘state’ of the system and the way it evolves with time – possibly towards thermal equilibrium.
Introduction to the density operator Motivation and definition It is often inconvenient, for one reason or another, to keep track of all the variables of a complex system. For example, in a many-body system it would be impracticable to consider the co-ordinates of each particle. Furthermore, such information is actually not of much interest. Thus in a spin system the behaviour of the individual magnetic moments is unimportant; the components of the total magnetisation are the variables of primary interest. Very generally, for a system with ∼ 10 23 co-ordinates, one is unlikely to need more than, say, ten variables to describe its observable properties. However, it is quite clear that this reduced amount of information is no longer sufficient to write down a wavefunction and, therefore, it is no longer possible to calculate the evolution of such a system using the usual methods of quantum mechanics. In Chapter 5 we saw that it was possible, by the introduction of probabilistic arguments, to calculate the evolution of the total magnetisation of a spin system. The calculations were performed using the machinery of quantum mechanics and meaningful and useful results were thereby obtained. In this chapter we will look at things from a rather different point of view. We will not, initially, be concerned with the expectation values of certain specified observables, rather we direct our attention to the general description of the ‘state’ of the system and the way it evolves with time – possibly towards thermal equilibrium.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
While the dynamic state of a single particle can be specified quantum mechanically in terms of its state function, any rigorous description of the state of a many-particle system would require the complete knowledge of the dynamic state functions of all the particles. That is not always possible. On the other hand, for a large number of particles in, or near, thermal equilibrium in a uniform sample, the principles of statistical mechanics may be invoked to describe the averaged expectation values of the physically observable properties of such a many-particle system. The basic concepts of the density-matrix formalism and the quantum mechanic analog of the classical Boltzmann equation commonly used for optical and magnetic resonance problems of many-particle quantum systems are introduced in this chapter. Applications of this approach to such specific problems as the resonant interaction of electromagnetic radiation with optical media of two-level atoms, nonlinear optics, and the laser rate equations and transient dynamics are discussed in this chapter. Definitions of the density operator and the density matrix Up to this point, in studying the dynamics of quantum mechanic systems, we have assumed that the state of the system can be specified in terms of a precisely known state function |Ψ〉. On the other hand, for a macroscopic medium containing many particles, it is not always possible to know the exact dynamic states of all the particles in the medium, even for physically identical particles.
Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .
Ensembles and expectation values Equations of motion provide predictions of time-evolving variables of a dynamical system, and thereby enable us to predict the changing behavior. These equations are characteristic of a particular type of system, say a classical harmonic oscillator or a two-state quantum system. To complete the description of behavior one must specify initial conditions that select, from the many possible solutions to the equations of motion, a particular realization. In the example of a classical point particle the equation is second order in time, and so we need two initial conditions, of position and velocity, to specify a particular solution. These two numbers, taken with the classical equation of motion, provide a complete description of the behavior of a specific harmonic oscillator. Similarly the two-state quantum system requires two initial values to complete its definition. The initial conditions are never defined with infinite precision, even for classical mechanics. More generally we deal with an ensemble, that is, a collection of similar dynamical systems (e.g. oscillators) that differ in their preparation. The individual cases share some common attributes, but not necessarily all. Thus we have only incomplete information about any single system. We might, for example, consider planetary motion in which we know only the mean energy and the orbital plane. The loci of positions of the planets then form ellipses that make up the ensemble. Although it is possible to prepare a single two-state quantum system in which the initial state is known (apart from an overall phase), often one deals with ensembles of such systems, each of which may differ in some uncontrollable property.
Up until this point, we have concerned ourselves with a description of quantum mechanics centered upon how a state |ψ〉 evolves in time. Using this we can show that the expectation value of an operator evolves as 〈A(t)〉 = 〈ψ(t)| ˆA|ψ(t)〉 (6.1) However, for many instances, especially if we are interested in describing relaxation processes, it is useful to introduce the density operator ρˆ(t) = |ψ(t)〉〈ψ(t)| (6.2) taken as the outer product of the state vector with itself. From this definition we can write ρˆ = ∑ mn c∗ncm |m〉〈n| (6.3) = ∑ mn ρmn|m〉〈n| (6.4) where the ρmn are the density matrix elements. Expectation values of operator are then given by the trace 〈A(t)〉 = ∑ mn Amnρmn = Tr [ ˆAρ(t)] (6.5) where Tr [ ˆAρ(t)] denotes the trace operation: Tr [ ˆAρ(t)] = ∑ nn′ If ρ is diagonal, then Tr [ ˆAρ(t)] = ∑ Annρnn = ∑ 〈n|A|n〉Pn where Pn = ρnn is the statistical probability of finding the system in state n. These statistical weights must be such that Pn ≤ 1 and∑ Pn = 1 Hence, we conclude that knowing ρ we can compute the statistical average of an operator A.