Free particle density matrix

The spin-free one-particle density matrix Fi = F is diagonal in the basis of the (spin-free) natural orbitals (NOs)... 

The electron density is closely related to a more general function, the so-called spin-free one-particle density matrix 6 7 8). Whereas the electron density is a function of the three coordinates x, y, z, the density matrix is a function of six coordinates, which are conventionally noted Xx,y, zi,xi,y, zx. In the case of a one-electron system, the density matrix is given by... 

The other main avenue to tackle the A-particle quantum problem with Pis focuses on the so-called pair product actions. These are approaches that have been proven to be very cost effective, as relatively low P discretizations work excellently well. Besides, the use of pair actions has led to a deep understanding of helium and the system of hard spheres at low temperatures [83-108]. Clear indications of how to proceed along this line were given by Barker in his pioneering work on quantum hard spheres (QHS) [24], Klemm and Storer [83], and Ceperley [28]. Essentially, this sort of approach consists of representing the V-particle density matrix through a product of density matrices, which includes those of the free particles and those of the reduced masses of every pair of particles. The potential is assumed pair-wise additive and, for definiteness, P time slices in... 

To evaluate the density matrix at high temperature, we return to the Bloch equation, which for a free particle (V(x) = 0) reads... 

Example 4. For a given lattice, a relationship is to be found between the lattice resistivity and temperature usiag the foUowiag variables mean free path F, the mass of electron Af, particle density A/, charge Planck s constant Boltzmann constant temperature 9, velocity and resistivity p. Suppose that length /, mass m time /, charge and temperature T are chosen as the reference dimensions. The dimensional matrix D of the variables is given by (eq. 55) ... 

In conclusion of this section, we write out the expressions for the density matrix of a free particle and a harmonic oscillator. In the former case p(x, x P) is a Gaussian with the half-width equal to the thermal de Broglie wavelength... 

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...

In this approach, the density matrix is evaluated self-consistently with both the large and small component spinors, cpf and tpf, which can be reconstructed from the free-particle FW spinors ij/, in the Schrodinger picture... 

The significance of this is that the solutions to the Bloch equation are known for certain simple systems. For example, for a free particle (F = 0) in a one-dimensional space we find from the Bloch equation that the density matrix is [66]... 

We follow Doll et al. [42] and begin by writing a Fourier path-integral (FPI) formula for the ratio of the density matrix of the full system to that of a free-particle system (zero potential energy). The Jacobian and other prefactors cancel in the ratio, and we obtain... 

Moreover, this term is the difference of the kinetic energy density of the actual system and of that of a system of spin-free independent particles both with identical one-particle densities />(r).For real wavefunctions or for stationary states, it is simply the difference of the definite positive kinetic energies since the (unwanted) remaining contributions cancel one another. Another attractive property of the non-von Weizsacker contribution is that it appears to be the trace of the Fisher s Information matrix[28]. 

Another set of fundamental properties of metal clusters involves their response to static external electric and magnetic fields. Transition metal clusters embedded in matrices have been extensively studied with these techniques. [143] Unfortunately, the size distribution of the particles is broad in these experiments and interactions with the matrix can introduce changes in the properties of the metal aggregates. This is also true for metal clusters stabilized by ligands, as will be discussed in Section 2.4.5.3. The study of clusters in molecular beams overcomes these difficulties. This powerful approach, when combined with mass spectrome-tric detection, allows the investigation of mass selected free clusters. The main disadvantage is that since the particle densities are quite low, most of the standard spectroscopic techniques cannot be used. 

In the previous chapters we saw that in so far as deep inelastic lepton-nucleon scattering was concerned the nucleon could be visualized as a bound system of constituent quark-partons, with which the lepton interacted as if they were free particles. Our aim now is to try to give some sort of justification for such a picture, and to derive more reliable results for deep inelastic scattering in which allowance is made for the internal longitudinal and transverse (Fermi) motion of the quark-partons. The approach also allows us to evaluate the forward hadronic matrix elements of currents which appear in the sum rules discussed in Chapters 16 and 17 in terms of parton number densities. 

The quantities lnd>(r (r)) are defined as the quotients (-Sp,lh), where is the so-called action for the problem under consideration and involves an integration of kinetic and potential contributions over the period 0dimensionless quantity - In (r" (t)), its relation to the product of the density matrix elements in Eqs. (14) and (16) being clear [28]. A few simple examples (e.g., free particle and harmonic oscillator) admit the exact application of the PI formahsm in the P t form [12, 13], but for general many-body quantum systems this is not possible. However, some analytic developments related to Eq. (15) have given rise to the so-called Feynman s semiclassical approaches, which will be considered in Section 111. To exploit the power of the PI formahsm computational schemes utilize finite-P discretizations. In this regard, given that approximations to calculate density matrix... 

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