Statistical mechanics wave function

The kind of statistics obeyed by the system depends on the symmetry properties of the quantum-mechanical wave functions describing the molecules composing the system [3-7], For example, in some cases the a values may be taken as either integers (0, 1,. . . ) or half-integers (, f,. . . ) the choice is based on the nature of the particular Schrodinger equation describing the molecule. 

The kind of statistics obeyed by the system depends on the symmetry properties of the quantum-mechanical wave functions describing the molecules composing the system [3-7]. 

M. Born (Edinburgh) fundamental research in quantum mechanics, especially for the statistical interpretation of the wave function. 

The key feature in statistical mechanics is the partition function Just as the wave function is the corner-stone of quantum mechanics (from that everything else can be calculated by applying proper operators), the partition function allows calculation of alt macroscopic functions in statistical mechanics. The partition function for a single molecule is usually denoted q and defined as a sum of exponential terms involving all possible quantum energy states Q is the partition function for N molecules. 

Another example of slight conceptual inaccuracy is given by the Wigner function(12) and Feynman path integral(13). Both are useful ways to look at the wave function. However, because of the prominence of classical particles in these concepts, they suggest the view that QM is a variant of statistical mechanics and that it is a theory built on top of NM. This is unfortunate, since one wants to convey the notion that NM can be recovered as an integral part of QM pertaining to for macroscopic systems. 

A theory for nonequilibrium quantum statistical mechanics can be developed using a time-dependent, Hermitian, Hamiltonian operator Hit). In the quantum case it is the wave functions [/ that are the microstates analogous to a point in phase space. The complex conjugate / plays the role of the conjugate point in phase space, since, according to Schrodinger, it has equal and opposite time derivative to v /. 

There is a close similarity with planar electromagnetic cavities (H.-J. Stockmann, 1999). The basic equations take the same form and, in particular, the Poynting vector is the analog of the quantum mechanical current. It is therefore possible to experimentally observe currents, nodal points and streamlines in microwave billiards (M. Barth et.al., 2002 Y.-H. Kim et.al., 2003). The microwave measurements have confirmed many of the predictions of the random Gaussian wave fields described above. For example wave function statistics, current flow and... 

For gas-phase molecules the assumption of electronic adiabaticity leads to the usual Bom-Oppenheimer approximation, in which the electronic wave function is optimized for fixed nuclei. For solutes, the situation is more complicated because there are two types of heavy-body motion, the solute nuclear coordinates, which are treated mechanically, and the solvent, which is treated statistically. The SCRF procedures correspond to optimizing the electronic wave function in the presence of fixed solute nuclei and for a statistical distribution of solvent coordinates, which in turn are in equilibrium with the average electronic structure. The treatment of the solvent as a dielectric material by the laws of classical electrostatics and the treatment of the electronic charge distribution of the solute by the square of its wave function correctly embodies the result of... 

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,... 

Just as there is a fundamental function that characterizes the microscopic system in quantum mechanics, i.e., the wave function, so too in statistical mechanics there is a fundamental function having equivalent status, and this is called the partition function. For the canonical ensemble, it is written as... 

Next take an ensemble of replicas of a system, distributed over wave functions i (v) with probabilities wv. The il/(v) may be any set of normalized functions, not necessarily orthogonal to one another. An observable A has in each if/(v) a quantum-mechanical expectation (1.2), and the statistical average over the ensemble is... 

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

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 ... 

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. 

At this stage, we are confident that a clear connection between Levy statistics and critical random events is established. We have also seen that non-Poisson renewal yields a class of GME with infinite memory, from within a perspective resting on trajectories with jumps that act as memory erasers. The non-Poisson and renewal character of these processes has two major effects. The former will be discussed in detail in Section XV, and the latter will be discussed in Section XVI. The first problem has to do with decoherence theory. As we shall see, decoherence theory denotes an approach avoiding the use of wave function collapses, with the supposedly equivalent adoption of quantum densities becoming diagonal in the pointer basis set. In Section XV we shall see that the decoherence theory is inadequate to derive non-Poisson renewal processes from quantum mechanics. In Section XVI we shall show that the non-Poisson renewal properties, revealed by the BQD experiments, rule out modulation as a possible approach to complexity. 

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