Quantum Partition Function Research Articles

Generalisations of the pitzer—gwinn and feynman—hibbs approximations for quantum partition functions

By using an expression of Feynman and Hibbs for the partition function Z = Tr[exp[- β( p 2/2 m + V)]) in terms of integrals over paths with fixed means, and substituting for the potential V a local harmonic approximant (following Miller), we obtain an improved Pitzer—Gwinn approximation, as simple to use as the original one, but applicable to a much broader class of potentials. Systematic corrections, in the form of cumulant and Wigner—Kirkwood expansions are then obtained.

Wigner-like expansion for the quantum-statistical mechanics of solids: Application to the sine-Gordon chain.

A semiclassical expansion for the quantum partition function of solids is presented. With respect to the usual Wigner method, the temperature range where the expansion is significant is broadened owing to the correct quantum-statistical treatment of the harmonic modes, while a Wigner-like expansion is retained only for the anharmonic part. As an application, the lowest quantum correction to the specific heat of a one-dimensional sine-Gordon model is calculated analytically.

Small ? asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials

To a gauge field on a principalG-bundleP→M is associated a sequence of quantum mechanical Hamiltonians, as Planck's constant ħ→0 and a sequence of representations π n ofG is taken. This paper studies the associated quantum partition functions, trace exp (−tH n ), and produces a complete asymptotic expansion, as ħ→0, ħ=1/n, of which the principal term, proportional to the classical partition function, is the familiar classical limit.

Rescaled coherent state theory of the λ-transition in liquid He4

Abstract The quantum partition function for the interacting boson system is calculated using a complete set of rescaled coherent states in which the helium atom occupation numbers are rescaled with a large scale factor. There are terms in the statistical weights that do not vanish in the limit of infinite scale factor. These scale independent terms are shown to be in the familiar form of the Ginzburg-Landau model, and are explained as the result of condensation.

Non-classical diffusion of lithium in lithium borate glasses

Abstract The electrical conductivity, σ as a function of the 6Li/7Li ratio in a series of (6Li/7Li)2O : 2·88 B2O3 glasses does not follow classical diffusion theory, in that the σ6/σ7 ratio is a function of temperature. The data are interpreted in terms of an absolute-reaction-rate theory incorporating quantum partition functions.

The classical limit of quantum partition functions

We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS 2 classical spins asL→∞] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has “diagonal form”).

Classical bounds on quantum partition functions

Explicit bounds on the quantum partition functions are given in terms of classical partition functions, incorporating effective pair potentials, which account for Fermi- and Bose-statistics, respectively. The bounds may be used for the limit ℏ→0 and eventually for showing the interchangeability of the classical with the thermodynamic limit. A simple derivation of the thermodynamic limit for free particles with general dispersions is given.

On the calculation of quantum partition functions and virial coefficients by path integration

A new method for calculation of the quantum second virial coefficient directly from the potential by a path integration technique is suggested. The separation of the so-called “bound part” of the second virial coefficient, which is of specific interest to chemical thermodynamics, is discussed. Three-dimensional Morse oscillator calculations are demonstrated. A comparison is made of the results obtained by path integration and by direct summation over vibration—rotation levels.

Densities of states and the complex partition function

A Fourier transform technique based on the method of Dubner and Abate is here applied to the energy-level density problem for quantum partition functions. It is shown that, for typical molecular systems at realistic energies, the integrated level count, W(E), is obtainable to near ‘spectroscopic’ accuracy and in reasonable computing time. A smooth Fourier resolution of the simple density function, g(E), is likewise possible. Though somewhat less economical, this procedure entirely removes the slight discrepancies found in the steepest-descent method and can reasonably be claimed to dispose once and for all of the problem of quantum level-density computations in molecular rate theory.

First-Order Phase Transitions in Quantum-Coulomb Plasmas

A simplified model is suggested for the understanding (in principle) of the mechanism of a phase transition in a Coulomb system on a uniform neutralizing background. Quantum theory is taken into account only in so far as it provides discrete collective energy levels and, possibly, fermion statistics for the calculation of the collective modes. Other features stemming from the uncertainty relations are supposed to be irrelevant to the mechanism of the phase transition. (This is qualitatively justified in the text which goes with Fig. 1.) This procedure provides an ``effective quantum Hamiltonian'' Hqu, which incorporates the relevant quantum features and from which one can calculate the (approximate) quantum partition function using classical methods: zapproxQu=1N!hN∫⋯∫ exp (−βHqu)d3x1⋯d3pN. In the evaluation of this integral we use the same approximation in which the plasma modes are collective, viz., the RPA (random phase approximation). Because of the freezing-out of the collective degrees of freedom at the relevant temperatures, and because the number of these degrees of freedom changes with temperature and density, Hqu describes a system with variable degrees of freedom, which seems to provide the mechanism for the phase transition. Preliminary numerical evaluations indicate a phase transition at T = 0 for an electron plasma at rs = 7.6, i.e., just below the region of metallic densities. This is shown by finding a concave region in the free energy as a function of the density. The ground is then prepared for numerical evaluation of the transition at T ≠ 0. For white dwarfs we find a transition temperature of 107 ∼ K°.

Isotope effects in phase equilibria; a new tool for the study of intermolecular forces

The renascence of interest in isotope effects in phase equilibria during the past decade has provided a new body of experimental data pertaining to intermolecular forces. Concurrent advances in the quantum theory of the liquid and solid states have led to the formulation of the mass dependence of the quantum partition function and its temperature dependence. The theory shows that in the first approximation the isotope effects in phase equilibria are determined by the mean value of the second derivative of the intermolecular potential. For monatomic systems a new set of molecular parameters have been derived for neon, krypton, and xenon based on the WHALLEY-SCHNEIDER parameters for argon. For the condensed rare gases, it has been shown that independent particle models, e.g. anharmonic Einstein model of the crystal and quantum cell models in the liquid, cannot account for the observed isotope effects. The isotope effect on melting is discussed in terms of the structure of the solid and the nearest neighbor numbers of the liquid.For polyatomic molecules, a wealth of new information and a number of new phenomena associated with intermolecular forces have been discovered. It is found that in the first approximation the intermolecular potential appears as the sum of atomic interactions. A development of this method leads to a direct and quantitative explanation of a number of hitherto incomprehensible phenomena. Isotopic substitution provides a direct demonstration of hindered rotation in the liquid. Two new types of quantum effects associated with molecular rotation have been found. The first type is a coupling of the molecular rotation with the molecular translation in the field of the intermolecular force. The effect persists even for free rotation and is most pronounced for molecules with small moments of inertia. The effect has been quantitatively investigated through the phase equilibrium behaviour of the isotopic and rotational forms of hydrogen. The second effect is a coupling of the molecular rotation with the molecular vibration in the field associated with the hindered rotation. A quantitative investigation of this rotation-vibration interaction has been made by the study of the vapor pressures of the deuteroethylenes. Studies of isotopic phase equilibria together with molecular spectroscopy of the condensed state provide information concerning the perturbation of the intramolecular vibrations by the intermolecular forces.

Smooth Curve Approximation to the Energy-Level Distribution for Quantum Harmonic Oscillators

It is pointed out that the distribution of energy levels for a set of quantum oscillators may be expressed exactly as the inverse Laplace transform of a function closely related to the quantum partition function. This observation leads quite easily to a smooth curve approximation for the energy-level distribution which can be expressed in terms of a generalized Bernoulli polynomial. A connection between this approximation and those given by Schlag and Sandsmark and by Whitten and Rabinovitch is established. A detailed comparison with Whitten and Rabinovitch's results is made for the special case of degenerate oscillators.

Quantum effects in the liquid state by means of a phenomenological cell model: The vapour pressure ratio of Ne and Ar isotopes

Quantum effects on the vapour pressure of argon and neon isotopes in the liquid state are evaluated by means of two semi-empirical cell models. The space arrangements of atoms around the cell is obtained through the experimental data on the radial distribution function; the energy levels of the oscillators, appearing in the quantum partition function, are calculated in the anharmonic approximation. At T = T tr the results are higher than the theoretical results for the solid, while they should be lower. This comparison points out the limits of lattice-cell models even at high densities, in spite of the fact that some agreement is found with the experimental data.