Exchange Effects in Quantum Statistical Mechanics

By Dr. Myron L. Cramer

A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) at the
University of Wisconsin, 1975.

Introduction

The quantum-mechanical principle of indistinguishability of identical particles is often interpreted with the idea that due to the overlapping of the individual particle wavefunctions, it is impossible to tell which function is due to which particle.

The principle of indistinguishability combined with the spin-statistics of the particles leads to exchange effects in observed quantities.

There are many examples of exchange effects in physics and chemistry. In nuclei, exchange effects of mesons lead to internuclear forces. On the atomic scale, the Pauli exclusion principle leads to the electron shell structure. Between atoms, short range interactions have repulsions due to distortions of atomic electron distributions. In metals, exchange interactions affect the electron distributions. The Heisenberg spin interaction between neighboring atoms in a ferromagnetic lattice is due to atomic overlap. Nuclei spin-statistics appear in the rotational spectra of diatomic molecules, both in molecular spectroscopy and in the specific heat of the gas. In liquid 3He and 4He, the different phases have different exchange structures. The superfluid phase of He suggest that to a large extent, 4He atoms can be treated as bosons.

We believe that exchange effects are also important between more massive objects like molecules, where the intermolecular forces are much weaker than those binding the molecules together. Here we regard the molecule as a composite boson or fermion. We consider some of the ways that exchange considerations affect the existence of molecular associates like the trimer.

This thesis is divided into two parts. Part I deals with exchange effects in a many particle system. Part II treats the exchange effects in the three-body problem.

We take the position that the wavefunction, even a trial wavefunction in a variational calculation, should carry the knowable information in the form of quantum numbers of the system.

There are two spin cases for a trimer composed of spin-1/2 fermions: the total spin can be 1/2 or 3/2. The exchange symmetries of these two fermion cases are different from each other and from the boson case. These different exchange symmetries lead to different forms of variational trial wavefunctions which lead to different variational energies for the three cases. From this we conclude that even on this relatively large scale, quantum indistinguishability is important as it may in certain cases restrict the formation of some of these molecular trimers.

Full Thesis

Exchange Effects in Quantum Statistical Mechanics

Journal of Chemical Physics

A portion of this research was published in the Journal of Chemical Physics, Volume 67 Number 4, August 15, 1977. JCP Article.