Quantum Bose-Einstein statistics

In the case of quantum mechanical particles that obey Bose statistics, any number of particles can be at each of the g, states associated with level i. We can think of the gi states as identical boxes in which we place a total of , particles, in which case the system is equivalent to one consisting of n, -I- gi objects which can be arranged in a total of 

---

It can be shown6 by applying quantum (Bose-Einstein) statistics and quantum (Schrodinger) waves to photons that... 

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

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

MaxweU-Boltzmaim 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 ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac 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). 

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. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

For a gas containing N molecules of the same chemical species, the molecules would all be indistinguishable from one another. The factor W has to be divided by Nl in this case. The proper explanation can only be understood through a detailed discussion of quantum mechanics and Bose-Einstein statistics. This explanation is beyond the realm of interest here, and we simply state the proper weighting for a collection of N indistinguishable molecules as... 

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. 

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 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-... 

There are various ways that fame comes to a scientist. For Satyendranath Bose it was asking Albert Einstein to run interference for him. Eventually his name was linked with Einstein s in both a statistical method of dealing with quantum particles, called Bose-Einstein statistics, as well as the peculiar state of matter known as the Bose-Einstein condensate. In addition, Bose had a class of particles named after him the boson. As this example illustrates, Einstein s scientific influence was telling. 

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. 

After the brilliant success of the Bose-Einstein statistics with the light quantum gas, it was a natural suggestion to try it in the kinetic theory of gases also, as a substitute for the Boltzmann statistics. The investigation, which was undertaken by Einstein (1925), is based on the hypothesis that the molecules of a gas are, like light quanta, indistinguishable from each other. 

We have shown in 4 (p. 209) that the introduction of the principle of indistinguishability into statistics leads to two, and only two, new systems of statistics, one of which, the Bose-Einstein statistics, we have discussed in detail in the last two sections (light quanta, gas molecules). We turn now to the second possible statistics, which is based on Pauli s principle, and was introduced by Fermi and Dirac. We have seen in 4 (p. 209) that this statistics is intimately connected with the employment of Pauli s principle, observing that the proper function of a state in which two electrons have the same partial proper function (with respect to the four quantum numbers, including the spin quantum number) automatically vanishes. 

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. 

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... 

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. 

There are other particles, called bosons, which satisfy other quantum statistics, the Bose-Einstein statistics, and that are described by wave functions symmetric with respect to the exchange, for which the Pauli principle is not valid. We may dispense with a further consideration of bosons in this chapter. 

---