Many-particle quantum system characterization

A many-particle quantum system is completely characterized by N and v(7>). Whereas % and rj measure the response of the system when N changes at fixed v(T), the polarizability (a) measures the response of the system for the variation of (v(r)) at fixed N when a weak electric field is the source of v(T), in addition to that arising out of a set of nuclei. Based on the inverse relationship [185] between a and rj, a minimum polarizability principle has been proposed the natural direction of evolution of any system is toward a state of minimum polarizability [186]. 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

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 mathematical basis of the relativistic quantum mechanical description of many-electron atoms and molecules is much less firm than that of the nonrelativistic counterpart, which is well understood. As we do not know of a covariant quantum mechanical equation of motion for a many-particle system (nuclei plus electrons), we rely on the Dirac equation for the quantum mechanical characterization of a free electron (positron) (Darwin 1928 Dirac 1928,1929 Dolbeault etal. 2000b Thaller 1992)... 

In conclusion, although the quantum nature of small systems is apparent, it only promotes the formation of stable structures of sizes beyond nanoscopic scales, if many particles cooperatively interact with each other. However, the amount of data that would be required to characterize this macrostate on a quantum level can neither be calculated because of its giant extent nor measured because of fundamental and experimental uncertainties. Thus, is it really necessary to go down to the quantum level to identify the system parameters which somehow contain the condensed information of the collective behavior of the individual quantum states Thus for, we have only talked about quantum fluctuations, but at nonzero temperatures, thermal fluctuations are also relevant. Thus, obviously, a theory that allows for the explanation of macroscopic phases and the transitions between these, i.e., a physical theory of everything [57], must be of statistical nature. 

In Chapters 1 through 4, we focused on a desaiption of matter at the molecular and atomic levels. In such a description, the state of the system is described quantum mechanically in terms of the wave function, which is a function of the positions of all the particles. However, without highly specialized equipment, the observable world is far removed from the molecular realm both in terms of the number of atoms or molecules ( 10 instead of just a few) and length scale (centimeters and meters instead of Angstroms and nanometers). Objects that are very large compared to the molecular scale are referred to as macroscopic It is both inconvenient and impossible to describe a macroscopic system in terms of the detailed atomic-scale variables of the constituent molecules—there are simply too many. Instead we characterize the state of macroscopic systems using a relatively small set of quantities, called macroscopic properties (or thermodynamic properties). Two important examples of such properties are pressure and temperature. 

For the time being, a rigorous characterization and modehng of particle interactions is only possible for simple model systems A nice and very recent example is the measurement of the second viral coefficient and structure factor of quantum dots in solution by three independent methods, namely small-angle X-ray scattering (SAXS), cryo-TEM, and centrifugation (Van Rijssel et al, 2014). Nevertheless, the above-described approach can be transferred to many technical applications in a qualitative and conceptual way. 

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