Density matrix quantum expression

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. 

In the quantum case this function is to be replaced by its quantum counterpart, the Wigner function [Feynman 1972 Garg et al. 1985 Dakhnovskii and Ovchinnikov 1985] expressed via the density matrix as... 

These new trajectories are the so-called reduced quantum trajectories [30], which are only explicitly related to the system reduced density matrix. The dynamics described by Equation 8.42 leads to the correct intensity (time evolution of which is described by Equation 8.40) when the statistics of a large number of particles are considered. Moreover, Equation 8.42 reduces to the well-known expression for the velocity held in Bohmian mechanics, when there is no interaction with the environment. 

By its size, this chapter fails to address the entire background of MQS and for more information, the reader is referred to several reviews that have been published on the topic. Also it could not address many related approaches, such as the density matrix similarity ideas of Ciosloswki and Fleischmann [79,80], the work of Leherte et al. [81-83] describing simplified alignment algorithms based on quantum similarity or the empirical procedure of Popelier et al. on using only a reduced number of points of the density function to express similarity [84-88]. It is worth noting that MQS is not restricted to the most commonly used electron density in position space. Many concepts and theoretical developments in the theory can be extended to momentum space where one deals with the three components of linear momentum... 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

For a pulse-type NMR experiment, the assumption has a straightforward interpretation, since the pulse applied at the moment zero breaks down the dynamic history of the spin system involved. The reasoning presented here, which leads to the equation of motion in the form of equation (72), bears some resemblance to Kaplan and Fraenkel s approach to the quantum-mechanical description of continuous-wave NMR. (39) The crucial point in our treatment is the introduction of the probabilities izUa which are expressed in terms of pseudo-first-order rate constants. This makes possible a definition of the mean density matrix pf of a molecule at the moment of its creation, even for complicated multi-reaction systems. The definition of the pf matrix makes unnecessary the distinction between intra- and inter-molecular spin exchange which has so far been employed in the literature. 

In the last equality here we have introduced the partial Wigner transforms of the density matrix and operator. The prime on the trace indicates a trace over the subsystem degrees of freedom. All information on the quantum initial distribution is contained in pw R,P, 0). In the evaluation of this expression we assume that the time evolution of Bw R, P,t) is given by Eq. (4). This... 

The quantitative characteristic of the alignment created is given, as already stated, by multipole moments of even rank. A more rigorous treatment of the expansion of the quantum mechanical density matrix over irreducible tensorial operators will be performed later, in Chapter 5 and in Appendix D. As an example we will write the zero, second and fourth rank polarization moments and [Pg.62]

Expressions which would be applicable for arbitrary angular momentum values can be obtained by using the quantum mechanical density matrix. 

Other forms of normalization, as well as forms denoting irreducible tensor operators may be found in [304] and in Appendix D. With the aid of the orthogonality relation one may easily express the quantum mechanical polarization moments fq and Pq through the elements of the density matrix /mm and... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

We now have a formula for constructing the density matrix for any system in terms of a set of basis functions, and from Eq. 11.6 we can determine the expectation value of any dynamical variable. However, the real value of the density matrix approach lies in its ability to describe coherent time-dependent processes, something that we could not do with steady-state quantum mechanics. We thus need an expression for the time evolution of the density matrix in terms of the Hamiltonian applicable to the spin system. 

Verify that the multiple quantum coherence expressed in the density matrix of Eq. 11.68 does not give rise to observable H magnetization. 

The general quantum chemical description of the nonlinear susceptibilities and hyperpolarizabilities in the density matrix formalism was developed by Bloembergen and Shen [36]. A simplification of this model for dipolar organic molecules, by only considering the transition between the ground state and the first excited state, led to the well-known two-state expression for the first hyperpolarizability [37] (Eq. (23)). 

While known quantum mechanical relationships require more information than is contained in just p(r) for the determination of the energy, only a portion of the information contained in the one-density matrix is required for this purpose (Bader 1980). The one-density matrix is a function of six variables. The physical information of F (r, r ) is however, contained in the neighbourhood of its diagonal elements r = r. To establish this, one expresses r >(r, r ) in a new system of coordinates defined by... 

Clearly, the fluctuation in the average population of a region 2 (denoted here by J ( 2)) also approaches zero when a single event dominates the distribution. A difficulty with using quantum probabilities to determine the extent of spatial localization is that their evaluation requires the use of the lull Mh-order density matrix. Because of this, the calculation of P ( 2) rapidly becomes prohibitive with increasing N. The fluctuation A(N, 2) however, can be expressed entirely in terms of the diagonal elements of just the second-order density matrix (Bader and Stephens 1974). 

Numerical evaluation of thermodynamic projserties using either the full quantum expressions for the partition function and the density matrix or the... 

The density matrix p R,P) is not stationary under quantum-classical dynamics. Instead, the equilibrium density of a quantum-classical system has to be determined by solving the equation itpwe = 0. An explicit solution of this equation has not been found although a recursive solution, obtained by expressing the density matrix pwe in a power series in ft or the mass ratio p, can be determined. While it is difficult to find the full solution to all orders in ft, the solution is known anal3dically to 0 h). When expressed in the adiabatic basis, the result is [5]... 

From eq. (15) it is clear that also the derivatives with respect to coordinates of nuclei of above expressions are required. The technique of evaluation of the hybrid coulombic and exchange integrals (18) and their derivatives can be found in the literature16,17 18. Matrix of the normal modes dRfin /5density matrix Pfiv are calculated by a standard quantum chemistry software for the evaluation of vibrational frequencies of molecules. 

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... 

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