Quantum Fermi-Dirac statistics

In the case of quantum mechanical particles that obey Fermi-Dirac statistics, each level i has occupation 0 or 1, so that the number of particles that can be accommodated in a level of energy e,- is , gi, where gi is the degeneracy of the level due to the existence of additional good quantum numbers. Since the particles are indistinguishable, there are 

In three-dimensional cartesian coordinates the wave-vector is k = kx + kyf + k z and the energy of the state with wave-vector k is 

We consider the particles enclosed in a box with sides 2Lx, 2Ly, 2L, in which case a relation similar to Eq. (G.46) holds for Akx, dfey, k.  

If the electrons have only kinetic energy, the total energy of the system will be 

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In the above-mentioned 1980 symposium (p. 8), the historians Hoddeson and Baym outline the development of the quantum-mechanical electron theory of metals from 1900 to 1928, most of it in the last two years of that period. The topic took off when Pauli, in 1926, examined the theory of paramagnetism in metals and proved, in a famous paper (Pauli 1926) that the observations of weak paramagnetism in various metals implied that metals obeyed Fermi-Dirac statistics - i.e., that the electrons in... 

As a simple illustration of Eq. (77), consider a system composed of three particles of the same energy that can occupy four quantum states. The case of Fermi-Dirac statistics is shown in Fig. 2a. 

The development of quantum theory, particularly of quantum mechanics, forced certain changes in statistical mechanics. In the development of the resulting quantum statistics, the phase space is divided into cells of volume hf. where h is the Planck constant and / is the number of degrees of freedom. In considering the permutations of the molecules, it is recognized that the interchange of two identical particles does not lead to a new state. With these two new ideas, one arrives at the Bose-Einstein statistics. These statistics must be further modified for particles, such as electrons, to which the Pauli exclusion principle applies, and the Fermi-Dirac statistics follow. 

The particles with half-integer spin values must obey the so-called Fermi-Dirac statistics. The requirement that the wave function of the system consisting of particles of this sort must be antisymmetric with respect to the permutation of the coordinates of any two particles is the mathematical expression of this fact. An electron has half-integer spin, therefore, the total wave function of any atom or ion must be antisymmetric with respect to the permutation of the coordinates (or quantum numbers) of each pair of electrons. 

Systems containing symmetric wave function components are called Bose-Einstein systems (129) those having antisymmetric wave functions are called Fermi-Dirac systems (130,131). Systems in which all components are at a single quantum state are called Maxwell-Boltzmann systems (122). Further, a boson is a particle obeying Bose-Einstein statistics, a fermion is one obeying Fermi-Dirac statistics (132). 

Maxwell-Boltzmann particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles are indistinguishable. For example, individual electrons in a solid metal do not maintain positional proximity to specific atoms. These electrons obey Fermi-Dirac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

Many interesting and useful concepts follow from classical statistical con side rations (eg, the Boltzmann distribution law) and their later modifications to take into account quantum mechanical effects (Bose-Einstein and Fermi-Dirac statistics). These concepts are quite beyond the scope of the present article, and the reader should consult Refs 14 16. A brief excursion into this area is appropriate, however. A very useful concept is the so-called partition function, Z, which is defined as ... 

No other closed form solution of the quantum-mechanical absorption characteristic based on Fermi-Dirac Statistics (as presented in Table 5.5.10-1) could be found in the literature. A flurry of activity in the 1990 s introduced a variety of extensions to simple expressions aimed at curve fitting empirical data. 

The log-normal function is unique and does not deserve modification. It occupies a unique position in both botany and biology that is critically related to the processes involved in growth. There are four major classes of statistics of interest in vision. They are the normal, the log-normal, the Stefan-Boltzmann and the Fermi-Dirac statistics. The first is often spoken of as Gaussian Statistics. It relies on a totally random series of outcomes in a linear numerical space. Log-normal statistics rely on a totally random series of outcomes in a logarithmic space. This space is the logarithm of the linear space of Gaussian Statistics. The Stefan-Boltzmann class of statistics apply directly to totally random events constrained in their total energy. They explain the thermal radiation from a physical body. The Fermi-Dirac Statistics are also known as quantum-mechanical statistics. Fermi-Dirac Statistics represent totally random events constrained as to the amplitude of a specific outcome. While Fermi-... 

For a symmetrical (D ) diatomic or linear polyatomic molecule with two, or any even number, of identical nuclei having the nuclear spin quantum number (see Equation 1.47) I = n + where n is zero or an integer, exchange of any two which are equidistant from the centre of the molecule results in a change of sign of i/c which is then said to be antisymmetric to nuclear exchange. In addition the nuclei are said to be Fermi particles (or fermions) and obey Fermi Dirac statistics. However, if / = , p is symmetric to nuclear exchange and the nuclei are said to be Bose particles (or bosons) and obey Bose-Einstein statistics. 

In the Einstein-Bose statistics, there is no such clear physical way to find the revised law of collisions as in the Fermi-Dirac statistics. The law can be derived from the quantum theory but not in a simple enough way to describe here. In contrast to the Fermi-Dirac statistics, in which the presence of one molecule in a cell prevents another from entering the same cell, the situation with the Einstein-Bose statistics is that the presence of a molecule in a cell increases the probability that another one should enter the same cell. In fact, the number of molecules going into the fcth cell per second turns out to have a factor (1 + Nk), increasing linearly with the mean number Nk of molecules in that cell. Thus, the law of collisions for the Einstein-Bose statistics is just like Eq. (4.1), only with + signs replacing the — signs. In fact, we may write the law of collisions for both forms of statistics in the form... 

In writing equation (4.6), we have assumed that the nuclei can be treated as Dirac particles, that is, particles which are described by the Dirac equation and behave in the same way as electrons. This is a fairly desperate assumption because it suggests, for example, that all nuclei have a spin of 1 /2. This is clearly not correct a wide range of values, integral and half-integral, is observed in practice. Furthermore, nuclei with integral spins are bosons and do not even obey Fermi Dirac statistics. Despite this, if we proceed on the basis that the nuclei are Dirac particles but that most of them have anomalous spins, the resultant theory is not in disagreement with experiment. If the problem is treated by quantum electrodynamics, the approach can be shown to be justified provided that only terms of order (nuclear mass) 1 are retained. 

Fermi-Dirac statistics — Fermi-Dirac statistics are a consequence of the extension of the application of Pauli s exclusion principle, which states that no two electrons in an atom can be in the same quantum state, to an ensemble of electrons, i.e., that no two could have the same set of quantum numbers. Mathematically, in a set of indistinguishable particles, which occupy quantum states following the Pauli exclusion principle, the probability of occupancy for a state of energy E at thermal equilibrium is given by f(E) = —(A)—, where E is the... 

In Fermi-Dirac statistics each state can accommodate at most only two particles with opposed spins. In Bose-Einstein statistics, just as in the classical Maxwell-Boltzmann statistics, there is no limitation to the number of particles in a given state. In classical statistics the particles in the same state were assumed to be distinguishable one from the other. As this assumption has been shown in quantum theory to be incorrect the particles in the same state in Bose-Einstein quantum statistics are indistinguishable. Interchanges of two of the par-... 

Einstein never failed to acknowledge Bose as the initiator of quantum statistics. (Two years later, in 1926, quantum statistics was extended independently by Paul Dirac and Enrico Fermi in what is now called Fermi-Dirac statistics.) But Bose was unable to extract firom his work the physical significance that Einstein was able to bring to it. With both Bose-Einstein and Fermi-Dirac statistics on the table, it took a few months for physicists to recognize their applications. 

The particles that make up the material world all belong to one of two groups bosons, the social particles that can come together in the same quantum state, and fermions, the antisocial particles, each of which demands a quantum state for itself. The former obey Bose-Einstein statistics and the latter Fermi-Dirac statistics. 

I7e. Heat Capacities at High Temperatures.—Although the theoretical treatment of heat capacities requires the limiting high temperature value to be 3/2, i.e., 5.96 cal. deg. g. atom , experimental determinations have shown that with increasing temperature Cv increases still further. The increase is, however, gradual for example, tfie heat capacity of silver is 5.85 cal. deg. g. atom at 300° K and about 6.5 cal. deg. g. atom at 1300° K. This increase is attributed mainly to the relatively free electrons of the metal behaving as an electron gas. By the use of the special form of quantum statistics, viz., Fermi-Dirac statistics, applicable to electrons, the relationship... 

Similar principles apply to ortho- and para-deuterium except that, as the nuclear spin quantum number of the deuteron is 1 rather than as for the proton, the system is described by Bose-Einstein statistics rather than the more familiar Fermi-Dirac statistics. For this reason, the stable low-temperature form is ort/io-deuterium and at high temperatures the statistical weights are 6 ortho 3 para leading to an upper equilibrium concentration of 33.3% para-deuterium above about 190 K as shown in Fig. 3.1. Tritium (spin i) resembles H2 rather than D2. 

The equations of quantum statistical mechanics for a system of non-identical particles, for which all solutions of the wave equations are accepted, are closely analogous to the equations of classical statistical mechanics (Boltzmann statistics). The quantum statistics resulting from the acceptance of only antisymmetric wave functions is considerably different. This statistics, called Fermi-Dirac statistics, applies to many problems, such as the Pauli-Sommerfeld treatment of metallic electrons and the Thomas-Fermi treatment of many-electron atoms. The statistics corresponding to the acceptance of only the completely symmetric wave functions is called the Bose-Einstein statistics. These statistics will be briefly discussed in Section 49. 

As the research on centroid theory evolves in the future, additional far-reaching questions will probably arise. For example, can electronically nonadiabatic transitions be readily included in the CMD method Can Bose-Einstein and Fermi-Dirac statistics be included Correspondingly, is Bose condensation related to the coalescence of centroid momenta Can the Pauli exclusion principle be described by effective repulsive or imaginary terms in the centroid potential to simulate real-time Fermion dynamics These are questions with unknown answers, but the path centroid perspective is clearly a different and promising way to describe complex quantum systems which should continue to yield new and interesting results for some time to come. 

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