Density operator formalism

A good introductory treatment of the density operator formalism and two-dimensional NMR spectroscopy, nice presentation of Redfield relaxation theory. 

The calculations presented here are based on the density operator formalism using the Liouville-von-Neumann equation and the theoretical approach is confined to quadrupolar nuclei subjected to EFG as well as CSA-interactions. Following the approach of Barbara et al.,20 the Hamiltonian for an N-site jump may be written as... 

For readers interested in the quantitative description of the time-dependent magnetization processes, the magnetization grating , that occurs in Mims ENDOR using the density operator formalism, which leads to the experimental behavior described earlier, the reader is referred to other sources by authors infinitely more qualified than the present... 

The state of an NMR-relevant physical system changes over time as described by the Schrodinger equation which within a statistical density operator formalism may be recast in form of the so-called Liouville-von Neuman equation... 

This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenberg-type uncertainty principles, the Schrodinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (r, p f) is now the quantum mechanical density operator (often referred to as the density matrix ), whose time evolution is detennined 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 diagrammatic approach that can unify the theory underlying these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the form of the time evolution of the light/matter density operator. (It is recommended that anyone interested in advanced study of this topic should familiarize themselves with density operator formalism [8, 9,10, JT and 12]. Most books on nonlinear optics [13. 14. 15. 16 and 17] and nonlinear optical spectroscopy [18. 19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be determined. The ensemble averaged electrical polarization, P, is then obtained— the centrepiece of all spectroscopies based on the electric component of the EM field. 

Analysis of the dynamic magnetic resonance experiments is achieved by using the density operator formalism, outlined elsewhere [10, 35, 49]. Here we summarize important features of this treatment and introduce the simulation paranKters. The spin Hamiltonion representing Zeeman, quadrupole or hyperfine interactions... 

The relaxation processes can be also described using the density operator formalism. To account for the longitudinal and transverse relaxations for a simple spin 1 /2 system in a phenomenological way [24], the relaxation effects on the time evolution of the density matrix, p t), can be estimated by ... 

One of the first fields where the quantum nature of the underlying system carmot be ignored, simply because there is no classical limit, is that of NMR. This is also a field where decay is directly observed in spin relaxation back to equilibrium. By giving a radio frequency pulse to an equilibrium system of spins, these are brought out of equilibrium, and, after the pulse, decay back to the equilibrium state. This free induction decay was first modeled phenomenologically by the Bloch equations, which pointed to the existence of two relaxation times, commonly called Tj and T2, but at a later stage Redfield used the density operator formalism to... 

There are a number of situations where quantum mechanics plays a role which can be clarified using the density operator formalism. Spectroscopic problems can, in general, be solved in a systematic way by a series expansion in the interaction with classical Hght fields, with a reasonable model - the Brownian oscillator model - for the line widths and Hne shapes of the transitions. Excitation with short pulses leads to other interesting aspects, as we showed in Section 9.15, which challenge the concept of reaction rates for those processes. But also in that case the lack of a good theory for quantum decay prohibits a clear discussion of the meaning of the rate of transfer. 

The density operator in the coordinate representation is given by the functions T q, Q, q Q, <), and can be expressed for Hamiltonian systems in terms of its amplitudes, which become the wavefunctions Q, <) I convenient to introduce the formally exact eikonal representation. 

Here 0 is the Heaviside function. The projection operator formalism must be carried out in matrix from and in this connection it is useful to define the orthogonal set of variables, k,uk,5k > where the entropy density is sk = ek — CvTrik with Cv the specific heat. In terms of these variables the linearized hydrodynamic equations take the form... 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

We now describe a generic formalism for active control of a molecule coupled to the radiation field. That is, we examine how the control conditions for a variety of circumstances can be expressed in terms of the phase of the external field and the phase of the relevant dynamical variables. For simplicity, we consider a simple case, namely, when only two electronic states of the molecule play roles in the reaction dynamics we take these to be the ground electronic state and the first excited electronic state. The radiation that couples the two surfaces is the means of control. The internal state of the molecule is defined by the density operators pj, j e g, e, where g and e denote the ground and excited states, respectively. The combined density operator describing the state of the system can be represented as... 

Thus the binary density operator is given only by the single-particle density operator Fl, which depends on the earlier time t0. Therefore, retardation effects appear. In order to eliminate this time we use the formal solution (1.30) for F, ... 

In order to construct a collision integral for a bound-state kinetic equation (kinetic equation for atoms, consisting of elementary particles), which accounts for the scattering between atoms and between atoms and free particles, it is necessary to determine the three-particle density operator in four-particle approximation. Four-particle collision approximation means that in the formal solution, for example, (1.30), for F 234 the integral term is neglected. Then we obtain the expression... 

Obviously we may expect that the simple two- and three-particle collision approximation discussed in the previous sections is not appropriate, because a large number of particles always interact simultaneously. Formally this approximation leads to divergencies. In the previous sections we used in a systematic way cluster expansions for the two- and three-particle density operator in order to include two-particle bound states and their relevant interaction in three- and four-particle clusters. In the framework of that consideration we started with the elementary particles (e, p) and their interactions. The bound states turned out to be special states, and, especially, scattering states were dealt with in a consistent manner. 

We present in Section 2 the formalism giving the equations for the reduced density operator and for competing instantaneous and delayed dissipation. Section 3 presents matrix equations in a form suitable for numerical work, and the details of the numerical procedure used to solve the integrodiffer-ential equations with the two types of dissipative processes. In Section 4 on applications to adsorbates, results are shown for quantum state populations versus time for the dissipative dynamics of CO/Cu(001). The fast electronic relaxation to the ground electronic state is shown first without the slow relaxation of the frustrated translation mode of CO vibrations, for comparison with previous work, and this is followed by results with both fast and slow relaxation. In Section 5 we comment on the general conclusions that can be reached in problems involving both vibrational and electronic relaxation at surfaces. 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

The density matrix representation is actually simpler than the product operator formalism for dealing with zero and multiple quantum coherences. Note that the type of multiple quantum coherence can be read from the lower left elements of the antidiagonal. ... 

A mode coupling theory is recently developed [135] which goes beyond the time-dependent density functional theory method. In this theory a projection operator formalism is used to derive an expression for the coupling vertex projecting the fluctuating transition frequency onto the subspace spanned by the product of the solvent self-density and solvent collective density modes. The theory has been applied to the case of nonpolar solvation dynamics of dense Lennard-Jones fluid. Also it has been extended to the case of solvation dynamics of the LJ fluid in the supercritical state [135],... 

We assume that the molecule is in a stationary state initially, the wave function of which is describable by HF. In the density matrix formalism [9, 10] (which is equivalent to the usual operator form), the Fock F(0) and density matrices D(Cl> satisfy the time-independent equation... 

A complete understanding of the processes involved in 2D NMR requires a more powerful theoretical underpinning than used in most of the book, so Chapter 11 is devoted to an introduction to the density matrix and product operator formalisms. These methods are not familiar to many chemists, but they are simple outgrowths of ordinary quantum mechanics. We examine the basic ideas and apply this theory in Chapters 11 and 12 to describe some of the most frequently used ID and 2D NMR experiments. 

Fortunately there is a simple mathematical formalism that gives us the best of the quantum and classical approaches. By recasting the time-dependent Schrodinger equation into a form using a so-called density operator, physicists have long been able to follow the development of a quantum system with time. This formalism... 

In Chapter 11 we shall also introduce the product operator formalism, in which the basic ideas of the density matrix are expressed in a simpler algebraic form that resembles the spin operators characteristic of the steady-state quantum mechanical approach. Although there are some limitations in this method, it is the general approach used to describe modern multidimensional NMR experiments. 

Chapter 11 Density Matrix and Product Operator Formalisms... 

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