Quantum statistical mechanics postulates

Note the qualitative — not merely quantitative — distinction between the thermodynamic (Boltzmann-distribution) probability discussed in Sect. 3.2. as opposed to the purely dynamic (quantum-mechanical) probability Pg discussed in this Sect. 3.3. Even if thermodynamically, exact attainment of 0 K and perfect verification [22] that precisely 0 K has been attained could be achieved for Subsystem B, the pure dynamics of quantum mechanics, specifically the energy-time uncertainty principle, seems to impose the requirement that infinite time must elapse first. [This distinction between thermodynamic probabilities as opposed to purely dynamic (quantum-mechanical) probabilities should not be confused with the distinction between the derivation of the thermodynamic Boltzmann distribution per se in classical as opposed to quantum statistical mechanics. The latter distinction, which we do not consider in this chapter, obtains largely owing to the postulate of random phases being required in quantum but not classical statistical mechanics [42,43].]... 

It is a fundamental postulate of quantum mechanics that the wave function contains all possible information about a system in a pure state at zero temperature, whereas at nonzero temperature this information is contained in the density matrix of quantum statistical mechanics. Normally, this is much more information that one can handle for a system with N = 100 particles the many-body wave function is an extremely complicated function of 300 spatial and 100 spin17 variables that would be impossible to manipulate algebraically or to extract any information from, even if it were possible to calculate it in the first place. For this reason one searches for less complicated objects to formulate the theory. Such objects should contain the experimentally relevant information, such as energies, densities, etc., but do not need to contain explicit information about the coordinates of every single particle. One class of such objects are Green s functions, which are described in the next subsection, and another are reduced density matrices, described in the subsection 3.5.2. Their relation to the wave function and the density is summarized in Fig. 1. 

To be more definite, the mass action law is a postulate in the phenomenological theory of chemical reaction kinetics. In the golden age of the quantum, chemistry seemed to be reducible to (micro)physics The underlying physical laws for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the diflBculty is only that exact application of these laws leads to equations much too complicated to be soluble (Dirac, 1929). As was clearly shown by Golden (1969) the treatment of chemical reactions needs additional requirements, even at the level of quantum statistical mechanics. The broad-minded book of Primas (1983), in which the author deeply analyses why chemistry cannot be reduced to quantum mechanics is strongly recommended. 

It is a fundamental postulate of quantum mechanics that the wave function contains all possible information about a system in a pure state at zero temperature, whereas at nonzero temperature, or in a general mixed state, this information is contained in the density matrix of quantum statistical mechanics. Normally, this is much more information that one can handle for a system with W = 100 particles the many-body wave function is an extremely complicated function of 300 spatial and 100 spin variables that would be impossible... 

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the ffee-electron gas and Bose-Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. 

The index of measurement statistics corresponding to a given preparation can be expressed in the form of a density operator 0. Some preparations result in states described by density operators that are pure (density matrices are idempotent), and some in states described by density operators that are mixed (density matrices are not Idempotent). In the context of the quantum mechanical postulates, the preceding sentence is all that need be said about any given preparation and, therefore, any given state. 

As reactants transform to products in a chemical reaction, reactant bonds are broken and reformed for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postulates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in unimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be formulated at the elementary microcanonical level and then averaged to obtain the canonical model. 

The measurement process is one of the most controversial areas in quantum mechanics. Just how and at what stage in the measurement process reduction occurs is unclear. Some physicists take the reduction of as an additional quantum-mechanical postulate, while others claim it is a theorem derivable from the other postulates. Some physicists reject the idea of reduction [see M. Jammer, The Philosophy of Quantum Mechanics, ley, 1974, Section 11.4 L. E. Ballentine, Am. J. Phys, 55, 785 (1987)]. Ballentine advocates Einstein s statistical-ensemble interpretation of quantum mechanics, in which the wave function does not describe the state of a single system (as in the orthodox interpretation) but gives a statistical description of a collection of a large number of systems each prepared in the same way (an ensemble) in this interpretation, the need for reduction of the wave function does not occur. [See L. E. Ballentine, Am. J. Phys, 40,1763 (1972) Rev. Mod. Phys, 42,358 (1970).] There are many serious problems with the statistical-ensemble interpretation [see Whitaker, pp. 213-217 D. Home and M. A. B. Whitaker, Phys Rep., 210,223 (1992) Problem 10.3], and this interpretation has been largely rejected. 

George Eugene Uhlenbeck (1900-1988). Dutch-American physicist. Born in Indonesia (then a Dutch colony), Uhlenbeck studied at the University of Leiden with Paul Ehrenfest, where he, with fellow student Samuel Goudsmit, postulated the existence of intrinsic electron spin. In addition to his work on quantum mechanics, Uhlenbeck made fundamental advances in statistical mechanics and the theory of random processes. 

Chapter 10 gives two further quantum-mechanical postulates that deal with spin and the spin-statistics theorem. 

First of all it is shown that the laws of thermodynamics are a consequence of the postulates of the quantum theory, together with one other postulate which is of a statistical character. Secondly statistical mechanics makes it possible to obtain important new theorems not known in pure thermodynamics, including methods of calculating heat capacities, free energies, etc., from spectroscopic data. 

A more complete version of the second postulate assumes random a priori phases of the time-dependent factors in the wave functions and involves quantum-mechanical density matrices. We present only the simplest version of statistical mechanics. 

An underlying idea in the statistical mechanical analysis of chemical systems is that all quantum states with the same energy are equally probable. For any one molecule, or any one quantum mechanical system, there is no a priori reason to favor one state of a given energy over another. This is a postulate we take it to hold so long as there are no violations in the predictions that follow from it. This idea was invoked in Chapter 1 in the statistical analysis that lead to the Maxwell-Boltzmann distribution law (Equation 1.11), and in Chapter 9 we found one direct experimental confirmation of the distribution law in the... 

Sometimes classical mechanics offers a sufficient description of the dynamical behavior of molecular sysfems. Such a description, however, does not provide the quantum energy levels that are involved in the postulate of equal probabilities. To apply in a classical mechanical framework, fhe postulate of statistical mechanics requires something analogous to quantum states and their energies. We consider the analogy to show that statistical mechanics can be applied without a quantum mechanical analysis however, the primary focus of this chapter uses quantum knowledge about molecules. 

In the standard theory of quantum mechanics, two kinds of evolution processes are introduced, which are qualitatively different from each other. One is the spontaneous process, which is a reactive (unitary) dynamical process and is described by the Heisenberg or Schrodinger equation in an equivalent manner. The other is the measurement process, which is irreversible and described by the von Neumann projection postulate [26], which is the rigorous mathematical form of the reduction of the wave packet principle. The former process is deterministic and is uniquely described, while the latter process is essentially probabilistic and implies the statistical nature of quantum mechanics. 

Several theorems that can be derived from the three postulates of quantum mechanics named above have been presented In the literature. One of these is that to every state of a system specified by means of a given preparation there corresponds a Her-mitian operator (3, called the density operator, which is an index of measurement statistics. The incorporation of the stable-equilibrium postulate into the theory, however, gives rise to additional theorems that are new to quantum physics. Some of these new theorems are as follows ... 

If we are again moving with the velocity v, then a Gaufi distribution will arise that broadens in time. Often this behavior is addressed that the particle flows away or melts in time. Thus, we are finding in the stochastic mechanics a very similar situation to that in comparison to the quantum mechanics. The concepts of the position and the momentum must be revisited and replaced by a new concept. In stochastic mechanics, we can reasonably justify that a particle never can be observed in an isolated manner. Here an interaction with other particles in the neighborhood appears. This interaction cannot be described by a deterministic method. Therefore, we chose a statistic description of the motion. Quantum mechanics starts with certain postulates that cannot be justified in detail. The success of the method defends it, even when the vividness suffers. 

Quantum mechanical principles. Fundamental constants of the universe the speed of light, die Boltzmann constant, the Planck constant. The wave-particle duality. The link between the Microscopic World of Energetic of Atoms/Molecules and the Macroscopic World de Broglie relationship, the Heisenberg relationships, and statistical distributions. The Bohr interpretation of the hydrogen atom. The postulates of quantum in the wave funetion. 

A specification of the state of a system in complete detail is therefore unattainable. This is one of the bases of the second law, as discussed already in 1 18, and it is for the same reason that it is impossible to calculate the average properties of a i stem by seeking to apply the laws of mechanics to the individual molectdes. In order to calculate this average U is necessary to use an extra postulate, over and above the postulates of mechanics. This postulate is of a statistical character, and it asserts that if there are Cl quantum states of an isolated system, all of them compatible with the fixed value of the energy, then the system is as likely to be found in any one of the states as in any other. Thus each of these quantum states is to be t For further diecussion on this point see Tolman, Prineiples of SiaHoHeal Mechanics (Oxford, 1938), 5, and Bom, Natural PhUoeophy of Cause and Chance (Oxford, 1949). 

Statistical methods are extensively employed at present. They make use of the postulates of statistical and quantum mechanics permitting, on the basis of the equipartition principle and of data concerning molecular structure, the calculation of the main energy contributions corresponding to individual types of motion. Structural data are obtained from spectra, recently predominantly from microwave spectra. Vibration frequency levels, which must be known if the contribution of vibrational motion is to be calculated, are likewise obtained from spectroscopic measurements. 

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