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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 20:16:08 +0000Phase space formulation of the quantum many-body problem
https://resolver.caltech.edu/CaltechTHESIS:11062012-103119433
Authors: Levine, Paul Hersh
Year: 1963
DOI: 10.7907/2JYR-A438
<p>By means of a quantum mechanical phase space distribution function
introduced by von Roos, the Schroedinger equation for a non-relativistic
system of N identical particles with scalar interactions is transformed
into a quantum mechanical generalization of the Liouville equation,
thereby formulating the problem in terms of a generalized density in
phase space, a quantity of primary interest in most treatments of the
corresponding classical system (or "plasma"). This transformation permits
a parallel development of the theories of classical and quantum plasmas
and thus allows the quantum many-body problem to be discussed virtually
completely in classical terms. In particular, a kinetic theory of
quantum plasmas is obtained by deriving the quantum analogue of the BBGKY
hierarchy, and applying thereto approximation techniques similar to those
of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from
similar previous studies based on the Wigner distribution function is that
the proper exchange symmetry can be tractably introduced into the formalism.</p>
<p>Attention is first focused on the Hartree and Hartree-Fock approximations,
in which case the quantum BBGKY system reduces to a simple quantum
generalization of the Vlasov equation. This equation is used to study
the response of spatially homogeneous systems to weak external forces,
and the associated problems of plasmon and spin-wave excitations. It is
also used to derive the quantum and exchange corrected equations of
inviscid hydrodynamical transport which are then applied to the problem
of sound propagation in the degenerate electron gas.</p>
<p>The second part of the study is concerned with the theory of the
many-electron atom in the Hartree and Hartree-Fock approximations.
The relevant quantum Vlasov equations lead naturally to a "statistical"
theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi
models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and
exchange corrections to these models are simultaneously generated. The
quantum hydrodynamical theory developed earlier is used to determine
the influence of these corrections on the boundary conditions of the
model, and a theory of the compressed atom is consequently obtained.
Considered in somewhat less detail are the effects of non-zero temperature,
net orbital angular momentum, relativity and correlations, as well as
time dependent processes.</p>
<p>The final part deals with the problem of the degenerate electron
gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair correlation functions for particles with
"parallel" and "anti-parallel" spin are obtained by neglecting three particle
correlations. From these functions, a quantum-mechanical collision
integral is derived which differs from that obtained by Silin
and Guernsey and conjectured by Wyld and Pines in that dynamical exchange
effects are included. Also obtained from the pair correlation function
is an expression for the "correlation energy" which reduces in the high
density limit to the result of Gell-Mann and Brueckner. At intermediate
densities an additional term appears in the energy due to the screening
of the exchange interaction by the dielectric properties of the medium.
It is evaluated in the high density limit and found to be -0.151 r_s ln r_s
Ryd/electron in marked disagreement with the corresponding value obtained
by DuBois.</p>
https://thesis.library.caltech.edu/id/eprint/7257