Dirac Einstein

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 Eermi-Dirac statistics. Eor this reason, the stable low-temperature form is orriio-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 190K as shown in Eig. 3.1. Tritium (spin 5) resembles H2 rather than D2. 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

Given the total density from Eq. (4.17), the temperature follows from the equation of state which depends in turn on what particles are present. For any one species i, with temperature T,. we have from the Fermi-Dirac or Bose-Einstein distribution, Eq. (2.41),... 

This topic is relevant to the physics of neutron stars (nuclei or quark bubbles embedded in a neutron gas), to dilute Bose-Einstein-condensate bubbles inside the background of a Fermi-Dirac condensate, to buckyballs in liquid mercury and to superconducting droplets in a Fermi liquid. 

As mentioned above, we assume that the molecular energy does not depend on the nuclear spin state For the initial rovibronic state nuclear spin functions available, for which the product function 4 i) in equation (2) is an allowed complete internal state for the molecule in question, because it obeys Fermi-Dirac statistics by permutations of identical fermion nuclei, and Bose-Einstein statistics by permutations of identical boson nuclei (see Chapter 8 in Ref. [3]). By necessity [3], the same nuclear spin functions can be combined with the final rovibronic state form allowed complete... 

Abraham Pais was a theoretical physicist and scientific historian, who died in 2000 at the age of 81. Pais worked with luminaries such as Einstein, Oppenheimer, Dirac, and Feynman. His best-known work was a biography of Einstein, Subtle Is the Lord The Science and the Life of Albert Einstein, which was published in 1982 and won the 1983 American Book Award. Now that Pais is gone, I have lost track of Einstein s pipe. 

In conclusion we must mention that a necessary condition for the validity of Eq, (3), and consequently of other formulas derived from Eq. (3) is that Ni < 1 for the state (or states) of lowest energy and a fortiori for all other states, When this inequality does not hold. Boltzmann s distribution law must be replaced by a more general and more precise distribution law, either that of Fermi and Dirac or that of Bose and Einstein according to the nature of the molecules. See also Statistical Mechanics. 

The next great development in physics was again an outgrowth of Einstein s ideas. Dirac was not satisfied with the fact that early quantum mechanics did not fit into the framework of relativity theory, The velocities of electrons in ordinary atoms are so small compared to the speed of light that the neglect of relativity theory did not matter much. Rut what about wave mechanics of particles that move much faster Dirac was able in 1927 to unite relativity with quantum mechanics. 

A few years later the antielectron was found, and almost 30 years later, the antiproton. Antimatter indeed exists in nature, as Dirac predicted from Einstein s work. This theoretical prediction was one of the greatest intellectual achievements of science. Today, beams of antimatter are produced in many laboratories they run in carefully evacuated tubes m order not to hit any ordinary matter until they reach their target, where they annihilate with the target substance. 

Also, at the aforementioned Pontifical Academy Session on Einstein, P.A M. Dirac observed ... 

Dirac, P.A.M. Einstein, Einstein Session of the Pontificial Academy, Vatican City (November 10, 1979). Reprinted in Science, 207, 1161-1162 (1980). 

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. 

Figure E.10 (a) Bose-Einstein distribution function, (b) Fermi-Dirac distribution function, and (c) filling of levels by fermions at T = 0 and T=T1>0. The dashed line indicates the Fermi energies p.

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). 

This attempt to incorporate the spin of the electron, by using a halfintegral quantum number in a theory which seems to require integral values appears very artificial. It does nevertheless agree with the experimental observations. In 1928, Dirac developed a theory of the electron wavefunction which incorporated the principles of Einstein s theory of relativity. Very remarkably, the spin appears as a natural prediction of that theory, although the mathematical details are much too complicated to discuss here. 

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 ... 

One of the simplest procedures to get the expression for the Fermi-Dirac (F-D) and the Bose-Einstein (B-E) distributions, is to apply the grand canonical ensemble methodology for a system of noninteracting indistinguishable particles, that is, fermions for the Fermi-Dirac distribution and bosons for the Bose-Einstein distribution. For these systems, the grand canonical partition function can be expressed as follows [12] ... 

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. 

PHYSICAL PRINCIPLES OFTHEQUANTUMTHEORY. Werner Heisenberg. Nobel Laureate discusses quantum theory, uncertainty, wave mechanics, work of Dirac, Schroedinger, Compton, Wilson, Einstein, etc. 184pp. 5X x 8)4. 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

Assemblies in the Molecular Phase Space.—When we describe a system by giving the Ni s, the numbers of molecules in each cell of the molecular phase space, we automatically avoid the difficulties described in the last section relating to the identity of molecules. We now meet immediately the distinction between the Fcrmi-Dirac, the Einstein-Bose, and the classical or Boltzmann statistics. In the Einstein-Bose statistics, the simplest form in theory wa art. up a complexion bv giving a set of Nj s, and we say that any possible set of Ni s, subject only to the obvious restriction... 

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