Many particle system density operator

In variational treatments of many-particle systems in the context of conventional quantum mechanics, these symmetry conditions are explicitly introduced, either in a direct constructive fashion or by resorting to projection operators. In the usual versions of density functional theory, however, little attention has b n payed to this problem. In our opinion, the basic question has to do with how to incorporate these symmetry conditions - which must be fulfilled by either an exact or approximate wavefunction - into the energy density functional. 

In addition to operators corresponding to each physically measurable quantity, quantum mechanics describes the state of the system in terms of a wavefunction F that is a function of the coordinates qj and of tune t. The function l F/(qj,t)l2 = P P gives the probability density for observing the coordinates at the values qj at time t. For a many-particle system such as the H2O molecule, the wavefunction depends on many coordinates. For the H2O example, it depends on the x, y, and z (or r,0, and < )) coordinates of the ten... 

Until now we assumed that we have the maximum information on the many-particle system. Now we will consider a large many-body system in the so-called thermodynamic limit (N- °o, V—> >, n = NIV finite) that means a macroscopic system. Because of the (unavoidable) interaction of the macroscopic many-particle system with the environment, the information of the microstate is not available, and the quantum-mechanical description is to be replaced by the quantum-statistical description. Thus, the state is characterized by the density operator p with the normalization... 

The quantum mechanical Lagrangian density for a system of many particles interacting via a many-particle potential energy operator V is... 

In a many-particle system, fluctuations of the density lead to scattering of radiation light, neutrons. X-rays. Density fluctuations lead to polarization fluctuations, which lead to scattering . One can define the density operator. 

A system of many identical particles is most conveniently described by the if) field operators (5). The generic form of the electronic charge density operator in a semiconductor is... 

First it is necessary to introduce appropriate notation and definitions and to recall some properties of the density operator introduced by von Neumann [10] and Dirac [11]. A general many-electron wavefunction, for a system in stationary state K, will be written where q stands for all the required particle variables (space and... 

This is a very useful expression for considering it associated with the mono-density operators when the many-fermionic systems are treated, although similar procedure applies for mixed (sample) states as well. There is immediate to see that for Af formally independent partitions the Hilbert space corresponding to the iV-mono-particle densities on pure states, we individually have, see Eqs. (4.176), (4.181), (4.182) and (4.184),... 

---