Fermi-Dirac and Bose-Einstein Statistics

If we are dealing with nondistinguishable particles, the number of quantum mechanical states is much lower than for distinguishable particles. For example, the lithium atom has three spin orbitals (Xis . Xisp. Xisa) One many-electron wave function is Tifl,2,3) = x,3 (l) x,3p(2) another is P2(1.2,3) = %i, (2) Xi,p(l)  

X2sc(3). Since electrons satisfy Fermi-Dirac (FD) statistics, however, there is only one possible wave function, the Slater determinant  

There is a single quantum mechanical state (in the case of the Li atom), not W = 3 = 6, as had been the case if the electrons had been distinguishable, or W = 3 /2 = 3 if the states had been considered spatial orbitals capable of accepting two electrons, according to Equation 5.2. 

Proceeding similarly if the wave function is the same (does not change sign) on interchange of two particle coordinates, we may write the wave functions that correspond to Equation 5.119  

Let us assume that we have three energy levels, a twofold degenerate lower one (for example, with two different spin) with energy E, = E2 = 0, and a higher one with energy E2 = AE. The number of possible states, b d Qbe if we first consider the particles as distinguishable in the Boltzmann case, are given in Table 5.2. 

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Equations (5.3.44) through (5.3.48) are the fundamental formulas of Fermi-Dirac and Bose-Einstein statistics. 

Since conditions encountered in combustion problems are virtually always in the range where departures from Boltzmann statistics are negligibly small, the results of Fermi-Dirac and Bose-Einstein statistics will not be considered here. Complications associated with strong many-body inter-molecular forces (for example, liquids) will also be neglected we shall restrict our attention to ideal gases, although many of the formulas will also be valid for solids. 

The subject of statistical mechanics is a branch of mechanics which has been found very useful in the discussion of the properties of complicated systems, such as a gas. In the following sections we shall give a brief discussion of the fundamental theorem of statistical quantum mechanics (Sec. 49a), its application to a simple system (Sec. 496), the Boltzmann distribution law (Sec. 49c), Fermi-Dirac and Bose-Einstein statistics (Sec. 49d), the rotational and vibrational energy of molecules (Sec. 49e), and the dielectric constant of a diatomic dipole gas (Sec. 49/). The discussion in these sections is mainly descriptive and elementary we have made no effort to carry through the difficult derivations or to enter into the refined arguments needed in a... 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

Box 2.6. Boltzmann, Fermi-Dirac and Bose-Einstein Statistics... 

The origin of the nonvauishing Joule-Thomson effect is the effective repulsive (Fermions) and attractive (Bosons) potential exerted on the gas molecules, which arises from the different ways in which quantum states can be occupied in sy.stems obeying Fermi-Dirac and Boso-Einstein statistics, respectively [17]. In other words, the effective fields are a consequence of whether Pauli s antisymmetry principle, which is relativistic in nature [207], is applicable. Thus, a weakly degenerate Fermi gas will always heat up ((5 < 0), whereas a weakly degenerate Bose gas will cool down (5 > 0) during a Joule-Thomson expansion. These conclusions remain valid even if the ideal quantum gas is treated relativistically, which is required to understand... 

Here, i and stand for initial states, k and I for final states and a i(gfi,4>) dQ, defined for a rearrangement collision in Eq. (161) has been substituted for the classical 2nb db. Equation (230) may be derived from the quantum mechanical Boltzmann equation of Uehling and Uhlenbeck with the assumption of Boltzmann statistics, i.e., that the low-temperature symmetry effects of Fermi-Dirac or Bose-Einstein statistics are negligible, but that the quantum mechanical collision cross-section should be retained. We should note that although one may heuristically introduce the quantum mechanical cross-section as in Eq. (230), the Uehling-Uhlenbeck or a similar equation is strictly derived for special interactions only. In this connection it is interesting that the method given later in Section V-C yields the same result [Eq. (330)] as that of this section [Eq. (251)] only when an approximation equivalent to the Bom approximation is made. 

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... 

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. 

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. 

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). 

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 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. 

In Eqs. (2.7) and (2.11), we have found the general expressions for the entropy in the Fermi-Dirac and Einstein-Bose statistics. From either one, we can find the entropy in the Boltzmann statistics by passing to the limit in which all N% s are very small compared to unity. For small N%, In (1 Nt) approaches iVt-, and (1 Nt) can be replaced by unity. Thus either Eq. (2.7) or (2.11) approaches... 

We now ask, how many collisions per second are there in which molecules in the ith and jth colls disappear and reappear in the fcth and Zth cells We can be sure that this number of collisions will be proportional both to the number of molecules in the ith and to the number of molecules in the jth cell. This is plain, since doubling the number of either type of molecule will give twice as many of the desired sort that can collide, and so will double the number of collisions per unit time. In the case of the Boltzmann statistics, which we first consider, the number of collisions will be independent of the number of molecules in the kth and Zth cells, though we shall find later that this is not the case with the Fermi-Dirac and Einstein-Bose statistics. We can then write the number of collisions of... 

The Kinetic Method for Fermi-Dirac and Einstein-Bose Statistics. The arguments of the preceding sections must be modified in only two ways to change from the Boltzmann statistics to the Fermi-Dirac or Einstein-Bose statistics. In the first place, the law giving the number of collisions per unit time, Eq. (1.1), must be changed. Secondly, as... 

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