The classic limit of quantum statistics

Using tliis result and suimning both sides of Eq. (B.71) over i, we find (Ai exp -A2Wj 0i = exp(-A2 /) a, a  

To prove Eq. (2.97) in the subsequent Appendix B.6.2 we first need to introduce a function  

Note that 6 x — b) is not really a function in the ordinary sense because 

Applying this logic to the integral on the right side of Eq. (B.78) we may write 

An interesting relation is obtained by considering the Fourier transform of / (r). By analogy with Eqs. (2.84) and (2.87) we may write 

Using tliis result and suuniiing both sides of Ekj. (B.71) over i, we find 

---

The link to the molecular level of description is provided by statistical thermodynamics whore our focus in Chapter 2 will be on specialized statistical physical ensembles designed spc cifically few capturing features that make confined fluids distinct among other soft condensed matter systems. We develop statistical thermodynamics from a quantum-mechanical femndation, which has at its core the existence of a discrete spectrum of energj eigenstates of the Hamiltonian operator. However, we quickly turn to the classic limit of (quantum) statistical thermodynamics. The classic limit provides an adequate framework for the subsequent discussion because of the region of thermodynamic state space in which most confined fluids exist. 

The path integral approach was introduced by Feynman in a seminal paper published in 1948. It provides an alternative formulation of time-dependent quantum mechanics, equivalent to that of Schrodinger. Since its inception, the path integral has found innumerable applications in many areas of physics and chemistry. Its main attractions can be summarized as follows the path integral formulation offers an ideal way of obtaining the classical limit of quantum mechanics it provides a unified description of quantum dynamics and equilibrium quantum statistical mechanics it avoids the use of wavefunctions and thus is often the only viable approach to many-body problems and it leads to powerful influence functional methods for studying the dynamics of a low-dimensional system coupled to a harmonic bath. 

In order to make the theory useful it is necessary to know the constant of proportionality, which is calculated in such a way as to give the classical limit of the number of quantum states. This matter is dealt with in standard books on statistical mechanics [26]. The result is that for a system with n degrees of freedom, i.e. n position coordinates q and n momentum coordinates p, the number of states in the infinitesimal volume element rfq rfp is equal to rfq rfp//i", where n is Planck s constant. The association of a phase space volume /i with each quantum state can be thought of as a consequence of the uncertainty principle, which limits the precision with which a phase point can be specified in a quantum mechanical system. 

In parallel there exist some attempts trying to introduce a field theory (FT) starting from the standard description in terms of phase space [4—6], Of course, the best way to derive a FT for classical systems should consist in taking the classical limit of a QFT in the same way as the so called classical statistical mechanics is in fact the classical limit of a quantum approach. This limit is not so trivial and the Planck constant as well as the symmetry of wave functions survive in the classical domain (see for instance [7]). Here, we adopt a more pragmatic approach, assuming the existence of a FT we work in the spirit of QFT. 

In section 1.2, we introduced the quantum mechanical partition function in the T, V, N ensemble. In most applications of statistical thermodynamics to problems in chemistry and biochemistry, the classical limit of the quantum mechanical partition function is used. In this section, we present the so-called classical canonical partition function. 

The three branches of quantum statistics (Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac) meld into one, known as the classic limiting case, if the following condition is met ... 

We have seen that decoherence theory, according to its advocates [128], makes the wave-function collapse assumption obsolete The environmental fluctuations are enough to destroy quantum mechanical coherence and generate statistical properties indistinguishable from those produced by genuine wave-function collapses. All this is unquestionable, and if a disagreement exists, it rests more on philosophy than on physical facts. Thus, there is apparently no need for a new theory. However, we have seen that all this implies the assumption that the environment produces white noise and that the system of interest, in the classical limit, produces ordinary diffusion. As we move from... 

At last, we can resolve the paradox between de Broglie waves and classical orbits, which started our discussion of indeterminacy. The indeterminacy principle places a fundamental limit on the precision with which the position and momentum of a particle can be known simultaneously. It has profound significance for how we think about the motion of particles. According to classical physics, the position and momentum are fully known simultaneously indeed, we must know both to describe the classical trajectory of a particle. The indeterminacy principle forces us to abandon the classical concepts of trajectory and orbit. The most detailed information we can possibly know is the statistical spread in position and momentum allowed by the indeterminacy principle. In quantum mechanics, we think not about particle trajectories, but rather about the probability distribution for finding the particle at a specific location. 

The key then is to somehow calculate the probability with which a specific quantum state contributes to the average values. As far as thermal systems in thermodynamic equilibrium are concerned, this is the central problem addressed by statistical thermodynamics. W( therefore begin our discuasion of some core elements of statistical thermodynamics at the quantum level but will eventually turn to the classic limit, because the phenomena addressed by this book occur under conditions where a classic description turns out to be adequate. We shall see this at the end of this chapter in Section 2.5 where we introduce a quantitative criterion for the adequacy of such a classic description. 

If another X group is added onto the aniline molecule, the An—X2 dissociation rate is more likely to be determined by the statistical dissociation step because with each additional nonlinear monomer the number of van der Waals modes increases by six. Because the van der Waals modes are extremely anharmonic and coupled to each other, a proper RRKM calculation should use anharmonic densities and sums. However, these are not yet generally available for the systems of interest. In all cases it is best to use the quantum density of states (i.e., RRKM) and not the classical approximation of it (RRK). With a binding energy of say 480 cm and six oscillators, the average energy per van der Waals mode is 60 cm. Since these frequencies typically vary between 20 and about 400 cm, it is evident that the average number of quanta excited per mode is only about 1 or 2, which does not correspond to the classical limit. 

It is important to look into the implications of Eq. (1) since the development of the quantum-statistical mechanical theory of Isotope chemistry from 1915 until 1973 centers about the generalization of this equation and the physical interpretation of the various terms in the generalized equations. According to Eq. (1) the difference in vapor pressures of Isotopes is a purely quantum mechanical phenomenon. The vapor pressure ratio approaches the classical limit, high temperature, as t . The mass dependence of the Isotope effect is 6M/M where 6M = M - M. Thus for a unit mass difference in atomic weights of Isotopes of an element, the vapor pressure isotope effect at the same reduced temperature (0/T) falls off as M 2. Interestingly the temperature dependence of In P /P is T 2 not 6X0/T where 6X.0 is the heat of vaporization of the heavy Isotope minus that of the light Isotope at absolute zero. In fact, it is the difference between 6, the difference in heats of vaporization at the temperature T from (> that leads to the T law. 

The microscopic description of a real gas considers a gas as a collection of identical quantum particles as part of the statistical approximation of the classical limiting case, i.e. as part of the approximation [7.61] which we have seen when establishing the partition function of molecules (see Chapter 5) ... 

Without any interaction and in the hypothesis of quantum statistics in the classical limiting case, relation [7.75] gives the canonical partition function, which we can write for a perfect pure gas in the form ... 

---