Density matrices momentum

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

Combining the inverses of (III. 14) and (III. 16) we get the natural expansion for a general element of the number density matrix in momentum space ... 

There are situations in which a definite wave function cannot be ascribed to a photon and hence cannot quantum-mechanically be described completely. One example is a photon that has previously been scattered by an electron. A wave function exists only for the combined electron-photon system whose expansion in terms of the free photon wave functions contains the electron wave functions. The simplest case is where the photon has a definite momentum, i.e. there exists a wave function, but the polarization state cannot be specified definitely, since the coefficients depend on parameters characterizing the other system. Such a photon state is referred to as a state of partial polarization. It can be described in terms of a density matrix... 

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... 

The electron and momentum densities are just marginal probability functions of the density matrix in the Wigner representation even though the latter, by the Heisenberg uncertainty principle, cannot be and is not a true joint position-momentum probability density. However, it is possible to project the Wigner density matrix onto a set of physically realizable states that optimally fulfill the uncertainty condition. One such representation is the Husimi function [122,133-135]. This seductive line of thought takes us too far away from the focus of this... 

R. Benesch and V. H. Smith, Jr., Density matrix methods in X-ray scattering and momentum space calculations, in Wave Mechanics—the First Fifty Years, W. C. Price, S. S. Chissick, and T. Ravensdale, eds. (Butterworths, London, 1973), pp. 357-377. 

To the extent that our model holds true, one can use the sum of the expressions (9.39) in the case of the large strata of the atom on hand for the density matrix of the atom in momentum space. But the knowledge of the density matrix allows one — as f)irac especially has pointed out—to answer all questions about the atom, in particular the calculation of the atom form factors. 

The essential information about transport properties in many-particle systems is given by the single-particle density matrix or by the singleparticle Wigner distribution. The equations of motion (1.18) and (1.23) for these important quantities are called kinetic equations. For the further consideration we write the latter equation in the momentum representation ... 

Using the relation, the binary density matrix in momentum representation may be expressed in terms of scattering wave functions. 

R. Benesch and S. R. Singh, Chem. Phys. Lett., 10, 151 (1971). On the Relationship of the X-Ray Form Factor to the First-Order Density Matrix in Momentum Space. 

The first term on the right-hand side is identical with that of Eq. (41) (since the nuclear kinetic energy cancel the Hamiltonian matrix Hrnn can be replaced by the PES matrix Vrnn, Eq. (10)). The derivatives in the second term on the right-hand side of Eq. (48) are responsible for the formation of a nuclear coordinate and momentum dependence of the density matrix. The multitude of involved coordinates and momenta, however, avoids any direct calculation of the pmn(R, / /,), and respective applications finally arrive at a computation of bundles of nuclear trajectories which try to sample the full density matrix. 

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-... 

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

The diagonal elements of the density matrix contain the populations of each of the BO states, whereas off-diagonal elements contain the relative phases of the BO states. The components of the density matrix with a = a describe the vibrational and rotational dynamics in the electronic state a, while the rotational dynamics within a vibronic state are described by the density matrix elements with a = a and va = v ,. The density matrix components with na = n a, describe the angular momentum polarization of the state Ja, often referred to as angular momentum orientation and alignment [40, 87-89]. The density matrix may be expanded in terms of multipole moments as ... 

One view of this trace operation is that the usual phase space integral may be obtained by representing the thermal density matrix e in plane-wave momentum states, and performing the trace in that state space (Landau et al, 1980, Section 33. Expansion in powers of h ). Particle distinguishabihty restrictions are essential physical requirements for that calculation. In this book we will confine ourselves to the Boltzmann-Gibbs case so that e = Q n, V, T)/n, since the... 

Throughout this section, the canonical density matrix and the Feynman propagator can be used interchangeably, the transformation P = it taking C into the propagator K, with t the time. While most frequently we shall use the coordinate representation r and r, it will be convenient in this section to work in k or momentum representation, by taking a double Fourier transform with respect to r and r. ... 

At the same time very often the real optical field interacting with atoms ha.s rather broad spectral profile, width of which is broader or comparable with the inhomogeneous width of the atomic transition. In this case, a broad spectral line approximation for quantum density matrix approach has proved to be verj- rewai d-ing. This approximation was introduced in the 1960s by C. Cohcn-Taimoudji for excitation of atoms with ordinai-y light sources [10]. This was an era before lasers. Later on it was adjusted for application for exedtation of atoms wdth multimode lasers [11] and for excitation of molecules in the case of large angular momentum states [3, 12]. 

If one is interested in a spatial distribution of angulcu momentum created by laser radiation, then there is a method how to make a transition from (juantum density matrix to the continuous angular momentum spatial distribution probability density. As it is shoAA n in [21], a classical probability density pci 6,if) for angulm momentum spatial distribution can be connected to the density matrix elements Imm At the J —> OC limit these elements can be considered as coefficients of the Fourier expansion of a classical probabihty densitj pd 9, p)... 

Here PMV denotes density-matrix elements, K = k0 - k, is the momentum transfer vector and Za is the nuclear charge of nucleous a. 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

Equation (41a) means that the function B( r) is equivalent to the volume integral of the density matrix y(ri, ri) under the condition of r = r - r, and Eq. (41b) means that B(r) is the autocorrelation function of the position wave function (r). The latter is an application of the Wiener-Khintchin theorem (Jennison, 1961 Bracewell, 1965 Champeney, 1973), which states that the Fourier transform of the power spectrum is equal to the autocorrelation function of a function. Equation (41c) implies not only that B(r) is simply the overlap integral of a wave function with itself separated by the distance r (Thulstrup, 1976 Weyrich et al., 1979), but also that the momentum density p(p) and the overlap integral S(r) are a pair of the Fourier transform. The one-dimensional distribution along the z axis, B(0, 0, z), for example, satisfies... 

According to Chaix et al. (1989), in the absence of external potentials ( A = V = 0) it has been argued that for stability we need a < 2/n and for instability we need a > 7r/log4. The stability result has been proved by Bach et al. (1999). To demonstrate the instability result a translationally invariant density matrix yc based on a rotation in momentum space has been employed (Chaix et al. 1989). But since the model is defined on R3, such a density matrix is either zero or has an infinite number of particles, it has either zero eneigy or infinite eneigy. This is because the state represented by the density matrix is the same in every unit volume. Therefore, the arguments presented by Chaix et al. (1989) could just show the instability within a unit volume. 

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