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Physical Review

Applications of Scattering Theory to Quantum Statistical Mechanics

Kenneth Watson

Published Jul 15, 1956• 112 citations
Publisher

Abstract

A Laplace transform relation between the distribution function $\mathrm{exp}{\ensuremath{-}\ensuremath{\beta}H}$ in quantum statistical mechanics and the Green's function of the wave equation has been noted by Koppe. This relationship may be extended to the stationary-state scattering equation. General properties of the transformation are discussed. Applications to the evaluation of the virial coefficients are considered. A further application of the perturbation methods of Brueckner and his collaborators is described.

DOI
10.1103/physrev.103.489

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A Laplace transform relation between the distribution function exp\ensuremath−\ensuremathβH\mathrm{exp}{\ensuremath{-}\ensuremath{\beta}H}exp\ensuremath−\ensuremathβH in quantum statistical mechanics and the Green's function of the wave equation has been noted by Koppe. This relationship may be extended to the stationary-state scattering equation. General properties of the transformation are discussed. Applications to the evaluation of the virial coefficients are considered. A further application of the perturbation methods of Brueckner and his collaborators is described.
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A formulation of quantum statistical mechanics is given, in terms of distribution functions. It is shown that all quantities of interest are obtained directly from nonlocal distribution functions ${\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta}), {\ensuremath{\gamma}}^{(2)}({{x}_{1}}^{\ensuremath{'}}, {{x}_{2}}^{\ensuremath{'}}, {x}_{1}{x}_{2}, \ensuremath{\beta}), \ensuremath{\cdots}.$ In a uniform system the Fourier transform of ${\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta})$ is the distribution of particles in kkk space, $n(k, \ensuremath{\beta})$. Various schemes for calculating these distribution functions directly are discussed. An approximate integral equation for $n(k, \ensuremath{\beta})$ is found, which can be solved by a converging iteration process. Some remarks are included on the application of the virial theorem.
Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.
The basic equations for the statistical system of waves in a randomly fluctuating medium are presented in the same form as in quantum mechanics, assuming a Markov process for the temporal change of the medium. A simple model is chosen for the Fokker-Planck equation of the medium which gives rise to a fluctuation of the Gaussian process, and several results are given to illustrate how the methods used in quantum mechanics or quantum field theory can be applied almost without change. Thus, the physical variables are represented by linear operators, and their equations of motion are determined by an equation similar to the Heisenberg equation of motion. The system has two stationary states (corresponding to the vacuum states 〈0∣〈0|〈0∣ and ∣0〉|0〉∣0〉 in quantum field theory), and the rather standard methods in field theory can be used for the evaluation of the Green's functions. When the temporal changes of the medium are sufficiently large compared with those of the waves, an adiabatic approximation is possible, and the isomorphic transformation (corresponding to the unitary transformation) is employed to lead to the result that the statistical system of the waves for this model is in perfect correspondence with the waves of bosons which interact with each other only through a two-body potential.
After a qualitative discussion of the advantages of the density matrix and of the different ways to introduce it (the statistical, quantum mechanical and operational methods of approach), § 2 deals with the general properties of the density matrix, including a discussion of pure cases and mixtures. A brief discussion is given of Green function techniques and of the relation between Green functions and correlation functions. A discussion of recent developments in the evaluation of partition functions concludes the first part of this article dealing with the theory of density matrix techniques. Sections 5 to 9 discuss applications. The first application is the quantum-chemical one to many-body systems in their ground state, that is, systems at absolute zero, and it is shown how the density matrix fits into the Hartree-Fock and Thomas-Fermi schemes. A brief discussion is given of the theory of diamagnetism. This is followed by a discussion of non-equilibrium processes and of Kubo's approach to transport theory. After that the polarization of beams of electrons or of photons is discussed and it is indicated how density matrix techniques can be used to treat scattering processes. Section 9 concludes this part of the paper by a brief account of density matrix theory applications to resonance and relaxation phenomena. Finally, the theory of measurement in quantum mechanics is considered.