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Royal Society of Chemistry eBooks

Lecture 9: The Density Operator

Giuseppe Pileio

Published Jun 13, 2022• 1 citation
Publisher

Abstract

In this lecture, I lay down the basics for the study of the dynamics of a large collection of identical spin systems by introducing the concept of a density operator and an equation for its evolution. Ensembles of spin systems represent macroscopic samples; in quantum mechanics, these can only be described in statistical terms through a density operator. This latter can be expressed in terms of spin angular momentum operators. The density operator allows all measurable properties of an ensemble of spin systems to be determined.

DOI
10.1039/9781837670871-00105

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In this lecture, I lay down the basics for the study of the dynamics of a large collection of identical spin systems by introducing the concept of a density operator and an equation for its evolution. Ensembles of spin systems represent macroscopic samples; in quantum mechanics, these can only be described in statistical terms through a density operator. This latter can be expressed in terms of spin angular momentum operators. The density operator allows all measurable properties of an ensemble of spin systems to be determined.
This chapter outlines the quantum ideas that underpin magnetic resonance and explains their link to classical physics. It discusses some of the essential concepts governing the quantum behaviour of ensembles of nuclear spins, including the nature of quantum states, the eigenvalue equation and observables, the measurement process, dynamics, and the Schroedinger equation. A detailed description of angular momentum is given along with discussion of the role of symmetry, rotation operators, and both product and total angular momentum representations. Statistical ensembles of spins are introduced and the density matrix and Liouville formalism outlined, along with the various tensor bases for the description of spin systems, both independent and coupled. The spin Hamiltonian is described in detail and the evolution of spin systems under various Hamiltonian terms is discussed using the density matrix description, along with suitable recession diagrams.
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.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
The density matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the density matrix). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations. They are introduced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T1 and T2, respectively, are also introduced and some numerical examples (results of Monte-Carlo simulations) are presented to illustrate their physical meaning.
The density matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the density matrix). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations. They are introduced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T1 and T2, respectively, are also introduced and some numerical examples (results of Monte-Carlo simulations) are presented to illustrate their physical meaning.