Density matrices quantum statistical mechanics

We begin with the theoretical and computational progress in solid-state NMR, which includes calculations of the lineshapes and dynamic processes based on density matrix theory or computations of the interaction parameters based on quantum (statistical) mechanics. 

To provide a definition of the density matrix in terms of fundamental wave-functions first consider the generalization of the expectation value from quantum mechanics to quantum statistical mechanics. In the quantum statistical case, an additional average over the probability density needs to be considered in the calculation of the expectation value ... 

It is a fundamental postulate of quantum mechanics that the wave function contains all possible information about a system in a pure state at zero temperature, whereas at nonzero temperature this information is contained in the density matrix of quantum statistical mechanics. Normally, this is much more information that one can handle for a system with N = 100 particles the many-body wave function is an extremely complicated function of 300 spatial and 100 spin17 variables that would be impossible to manipulate algebraically or to extract any information from, even if it were possible to calculate it in the first place. For this reason one searches for less complicated objects to formulate the theory. Such objects should contain the experimentally relevant information, such as energies, densities, etc., but do not need to contain explicit information about the coordinates of every single particle. One class of such objects are Green s functions, which are described in the next subsection, and another are reduced density matrices, described in the subsection 3.5.2. Their relation to the wave function and the density is summarized in Fig. 1. 

It is a fundamental postulate of quantum mechanics that the wave function contains all possible information about a system in a pure state at zero temperature, whereas at nonzero temperature, or in a general mixed state, this information is contained in the density matrix of quantum statistical mechanics. Normally, this is much more information that one can handle for a system with W = 100 particles the many-body wave function is an extremely complicated function of 300 spatial and 100 spin variables that would be impossible... 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

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. 

We have so far limited ourselves to a classical description, the natural requirement for which is the condition /, /" —> oo. In order that the description is valid for any angular momentum value, it is necessary to employ the quantum mechanical approach. We presume that the reader is acquainted with the density matrix (or the statistic operator) introduced into quantum mechanics for finding the mean values of the observables averaged over the particle ensemble. Under the conditions and symmetry of excitation considered here one must simply pass from the prob-... 

The crisis of the GME method is closely related to the crisis in the density matrix approach to wave-function collapse. We shall see that in the Poisson case the processes making the statistical density matrix become diagonal in the basis set of the measured variable and can be safely interpreted as generators of wave function collapse, thereby justifying the widely accepted conviction that quantum mechanics does not need either correction or generalization. In the non-Poisson case, this equivalence is lost, and, while the CTRW perspective yields correct results, no theoretical tool, based on density, exists yet to make the time evolution of a contracted Liouville equation, classical or quantum, reproduce them. 

The monomer label u is now interpreted as the Trotter (imaginary) time, and M as the mass of the quantum particle. The density matrix is relevant to the equilibrium statistical mechanics of a quantum particle, such as an electron in a dirty metal. 

So far we have not accomplished anything, except for a complicated rewrite of the Schrodinger equation. At this stage, the information content of both descriptions is exactly the same. However, the density matrix allows a description of quantum mechanical systems that is far broader than the wave function. To see that and why it is necessary, we move to quantum statistical physics. 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 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 detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

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